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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# transcendental num
bers.mws" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Title \+
" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 256 56 "An introduction \+
to the history of transcendental numbers" }{TEXT 298 61 "\n(and some r
elated Diophantine equations and approximations)\n" }{TEXT 257 1 "\n"
}{TEXT 297 58 "(in Maple mws or html formats at: www.spd.dcu.ie/johnbc
os)" }{TEXT 261 2 "\n\n" }{TEXT 258 7 "A talk " }{TEXT 749 14 "partly \+
covered" }{TEXT 750 83 " at the Dublin Branch of the Irish \nMathemati
cs Teachers Association, May 6th, 2004" }{TEXT -1 2 "\n\n" }{TEXT 259
0 "" }{TEXT 260 15 "For Fred Piper " }{TEXT 263 110 "(who taught me as
an undergraduate at Royal Holloway \nCollege, London Univ.) - with af
fection and admiration -" }{TEXT 262 19 " on his retirement\n" }{TEXT
322 61 "(From Fred Piper I first heard of Alan Baker, in Autumn 1967)
" }{TEXT 321 1 "\n" }{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 92 "Jo
hn Cosgrave, Mathematics Department,\nSt. Patrick's College, Drumcondr
a, Dublin 9, IRELAND." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "My aim"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Fields \+
Medallist W. T. Gowers in his essay " }{TEXT 903 29 "THE IMPORTANCE OF
MATHEMATICS" }{TEXT -1 94 " (see References) - the keynote address at
a Clay-funded millennium meeting in Paris - wrote: " }{TEXT 905 239 "
The percentage of the world's population, or even of the world's unive
rsity-educated polulation, who could accurately state a single mathema
tical theorem proved in the past fifty years, is small, and smaller st
ill if Fermat's last theorem " }{TEXT -1 24 "[proved by Andrew Wiles]
" }{TEXT 907 13 " is excluded." }}{PARA 0 "" 0 "" {TEXT -1 111 " W.
T.G.'s comment echoes a rebuke of Robert L. Devaney to the mathematica
l community. In the Preface to his " }{TEXT 906 29 "Chaos, Fractals, a
nd Dynamics" }{TEXT -1 1 " " }{TEXT 908 37 "(Computer Experiments in M
athematics)" }{TEXT -1 16 " Devaney wrote: " }{TEXT 909 452 "There are
a number of reasons why I have written this book. By far the most imp
ortant is my conviction that we in mathematics education - whether on \+
the secondary or collegiate level - fail miserably to communicate the \+
vitality of contemporary mathematics to out students. Let's face it: m
ost high school and college mathematics courses emphasize centuries-ol
d mathematics. As a consequence, many students think that mathematics \+
is a dead discipline..." }}{PARA 0 "" 0 "" {TEXT -1 248 " My modest
aim this evening is that all in my audience, besides coming to know s
omething of the origins and subsequent development of the theory of tr
anscendental numbers (and the related fields of Diophantine approximat
ions and equations), can " }{TEXT 911 77 "accurately state several mat
hematical theorems proved in the past fifty years" }{TEXT -1 171 " (po
ssibly previously unknown to them; don't worry, I won't be giving a te
st!), and, more importantly, have some notion with respect to the sign
ificance of those theorems." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "D
an Brown's " }{TEXT 833 17 "The DA VINCI CODE" }}{PARA 0 "" 0 ""
{TEXT -1 70 "Many of you may have read Dan Brown's 'international best
seller', The " }{TEXT 912 13 "DA VINCI CODE" }{TEXT -1 93 ". It's an u
nputdownable potboiler, which is currently being made into a film (I g
rew up with " }{TEXT 834 4 "film" }{TEXT -1 6 ", not " }{TEXT 835 5 "m
ovie" }{TEXT -1 194 "). Unfortunately it contains some erroneous Mathe
matics, on which I comment in a later section. Some teachers might wis
h to exploit the book's mathematical content as a basis for classroom \+
work." }}{PARA 0 "" 0 "" {TEXT -1 92 " One of the main characters i
n Brown's book is (French Agent) Sophie Neveu. Some quotes: " }}{PARA
0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 836 78 "Sophie Neveu ... who had stu
died cryptography in England at Royal Holloway... " }}{PARA 0 "" 0 ""
{TEXT -1 2 "2." }{TEXT 914 1 " " }{TEXT -1 0 "" }{TEXT 837 14 "'Your E
nglish " }{TEXT -1 9 "[S.N.'s] " }{TEXT 838 58 "is superb.' 'Thank you
, I studied at Royal Holloway.' ... " }}{PARA 0 "" 0 "" {TEXT -1 3 "3.
" }{TEXT 913 115 "'There's an easier way,' Sophie said, ' ... reflect
ional substitution ciphers ... A little trick I learned at Royal" }}
{PARA 0 "" 0 "" {TEXT 916 15 " Holloway.'" }{TEXT -1 3 " .." }
{TEXT 839 11 ". eyed her " }{TEXT -1 9 "[S.N.'s] " }{TEXT 840 78 "hand
iwork and chuckled. 'Right you are. Glad to see those boys at the Holl
oway" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 915 23 " are doing th
eir job.' " }}{PARA 0 "" 0 "" {TEXT -1 461 "That Sophie Neveu could ha
ve studied cryptography at Royal Holloway College in not a fiction, an
d indeed Fred Piper - former head of the Mathematics Department there \+
- is the about-to-retire Director of Royal Holloway College's Informat
ion Security Group. One of our own recent (2001) St. Patrick's College
BEd graduates studied at Royal Holloway for her Masters in Informatio
n Security (2002), and is still there, in the second year of her studi
es for a PhD. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Some prelimin
ary comments" }}{PARA 0 "" 0 "" {TEXT -1 228 "In Jan 2004, when I conc
eived the idea of giving a transcendental numbers talk, I naively tho
ught I would cover X, Y, and Z. However as soon as I started to make a
sketch of what I would do, that sketch got longer, and longer, " }}
{PARA 0 "" 0 "" {TEXT -1 406 "and quickly it began to dawn on me that \+
it would end up like my (Austria) August 2001 'Fermat's little theorem
' homage talk - in which I also used Maple - marking the 400th anniver
sary of Fermat's birth. In Austria all that I managed was to skim the \+
surface of my prepared material, and invite interested readers to acce
ss the complete talk at my web site. On this occasion that too is what
I intend to do." }}{PARA 0 "" 0 "" {TEXT -1 17 " A word about " }
{TEXT 616 7 "history" }{TEXT -1 492 " (it's in my title after all!). O
f course I am not a historian of Mathematics, so you won't find schola
rly history here. I believe though that everyone would agree that the \+
(difficult!) subject of transcendence didn't really begin until 1844, \+
in the sense that no one proved the transcendence of any number until \+
that year. I do attempt to sketch the scene prior to 1844, possibly no
t accurately. For example, I had always believed it was Euler who intr
oduced the concept of a transcendental " }{TEXT 617 6 "number" }{TEXT
-1 33 ", as opposed to a transcendental " }{TEXT 618 8 "function" }
{TEXT -1 132 ". There is an immense difference in mathematical difficu
lty between those two: it is almost completely trivial to prove (e.g.)
that " }{XPPEDIT 18 0 "e^z := Sum(z^n/n!,n = 0 .. infinity);" "6#>)%
\"eG%\"zG-%$SumG6$*&)F&%\"nG\"\"\"-%*factorialG6#F,!\"\"/F,;\"\"!%)inf
inityG" }{TEXT -1 21 " is a transcendental " }{TEXT 620 8 "function" }
{TEXT -1 80 ", but an entirely different matter to prove (as Hermite f
irst did in 1873) that " }{TEXT 619 1 "e" }{TEXT -1 21 " is a transcen
dental " }{TEXT 621 6 "number" }{TEXT -1 243 ". However, while I was p
reparing this talk it was brought to my attention by some corresponden
ts (see the transcendental numbers corner of my web site) that possibl
y Leibnitz knew of the concept of a transcendental number, and specula
ted that " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 109 " could be tra
nscendental (of course it is one matter to speculate, but an entirely \+
different one to prove...)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT 343 21 "For whom am I writing" }{TEXT -1 2 "? " }{TEXT
894 69 "Essentially I am writing for secondary school teachers of Math
ematics" }{TEXT -1 158 ", some of whom might wish to adapt some of wha
t I have written for their pupils, or perhaps even direct them to this
work. I have included some material that " }{TEXT 344 5 "could" }
{TEXT -1 53 " be presented to school pupils, for example a way of " }
{TEXT 326 6 "seeing" }{TEXT -1 145 " that Liouville's decimal number i
s transcendental without appealing to his approximation theorem, or a \+
beautiful demonstration of the fact that " }{XPPEDIT 18 0 "Pi <> 22/7;
" "6#0%#PiG*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 60 ", which provides much
fodder for further guided speculation." }}{PARA 0 "" 0 "" {TEXT -1
218 " I hope, too, that even experts might find something of intere
st here, if only my C. A. Rogers, K. F. Roth, R. Rado, and T. Schneide
r anecdotes, or the reference to my corrected version of the Gelfond p
roof [in the " }{TEXT 280 15 "Gelfond-Linnik " }{TEXT -1 284 "book] of
the real case of Hilbert's seventh problem, which I included in my Ma
nchester university course on transcendental numbers in 1972-73. My Ma
nchester 1972-73 course had complete details on Liouville's 1844 trans
cendence proof, proofs of the irrationality of integral powers of " }
{TEXT 327 1 "e" }{TEXT -1 23 ", the transcendence of " }{TEXT 328 1 "e
" }{TEXT -1 33 " (Hermite), the irrationality of " }{XPPEDIT 18 0 "Pi;
" "6#%#PiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi^2;" "6#*$%#PiG\"\"#
" }{TEXT -1 23 ", the transcendence of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG
" }{TEXT -1 262 " (Lindemann), Gelfond's solution of Hilbert's seventh
problem, Gelfond's earlier solution of a special case of that problem
, etc. I also included in my Manchester lectures a much corrected vers
ion of the proof from the extraordinary book by Gelfond and Linnik - \+
" }{TEXT 895 44 "Elementary Methods in Analytic Number Theory" }{TEXT
-1 134 " - of the real case of Hilbert's seventh problem ('elementary'
in the classical sense of making no use whatever of Complex Analysis)
. " }}{PARA 0 "" 0 "" {TEXT -1 277 " Not all my Dublin audience, or
later web readers, will be Maple users, and, on the web, will be read
ing the html version of this active Maple worksheet, and thus I will o
ccasionally include some slightly tangential material to impress on no
n-users the power of using Maple." }}{PARA 0 "" 0 "" {TEXT -1 644 " \+
Reluctantly I have omitted much material (a novice reader should unde
rstand that I have only scratched the surface, and even then not deepl
y), and have even scrapped substantial material that I had begun (e.g.
Mahler's A-, S-, T-, U- classification of transcendental numbers, for
I could see no end to where I could draw an eventual line). I also be
gan a substantial section on Special Functions (exponential, trigonome
tric, elliptic, elliptic modular, gamma, beta, zeta), but have entirel
y removed it in an attempt to reduce the eventual size. Because I coul
d not count on a general reader knowing about Algebraic Number Theory,
nor about " }{TEXT 342 1 "p" }{TEXT -1 381 "-adic analysis, nor about
... then I have also omitted much further work: e.g. I have had to om
it all reference to Siegel's E-functions work, and to (e.g.) Shidlovsk
ii, and Feldman. I also made a reluctant decision not to mention any w
ork of any of those who got attracted into transcendence after Baker b
egan his mid 1960's revolution (if I were to start, where would I fini
sh?). " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 44 "Hardy and Wright, Lang
, Stolarsky, Dieudonn\351" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 5 "From " }{TEXT 323 14 "Hardy & Wright" }{TEXT -1 11
"'s classic " }{TEXT 276 21 "The Theory of Numbers" }{TEXT -1 71 " [my
1962 edition, which I bought, and read, while at school] I quote: " }
{TEXT 277 71 "It is not immediately obvious that there are any transce
ndental numbers" }{TEXT 751 3 "..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 18 "Hardy & Wright do " }{TEXT 278 3 "not" }
{TEXT -1 22 " give a casual reader " }{TEXT 279 3 "any" }{TEXT -1 53 "
idea as to why it is 'not obvious'. I hope to do so." }}{PARA 258 ""
0 "" {TEXT -1 10 "__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT 324 10 "Serge Lang" }{TEXT -1 53 " (1966, Introduction
to Transcendental Numbers) ... " }{TEXT 281 202 "it is remarkable th
at a mathematical theory as old as the theory of transcendental number
s (dating back to Hermite's first result of 1873, the transcendence of
e) is still in what can only be called an " }{TEXT 340 21 "under-deve
loped state" }{TEXT -1 2 " [" }{TEXT 338 5 "Baker" }{TEXT -1 123 " was
just about to change all of that; it's interesting - and unfair, I be
lieve - that Lang doesn't date back to Liouville]" }}{PARA 258 "" 0 "
" {TEXT -1 10 "__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 329 18 "Kenneth Stollarsky" }{TEXT -1 124
" (1978, Bulletin American Mathematical Society review of books on tra
nscendental numbers by Baker, Mahler, and Waldschmidt) " }{TEXT 330
76 "The last dozen years have been a golden age for transcendental num
ber theory" }{TEXT 339 202 ". It has scored successes on its own groun
d, while its methods have triumphed over problems in classical number \+
theory involving exponential sums, class numbers, and Diophantine equa
tions... The result " }{TEXT -1 27 "[of Gelfond and Schneider] " }
{TEXT 331 37 "has now been brilliantly extended by " }{TEXT 333 5 "Bak
er" }{TEXT 334 67 " in several useful ways... At present the high poin
t of the theory " }{TEXT -1 86 "[of work that began with Liouville, an
d then was revolutionised by Thue, Siegel, ...] " }{TEXT 332 39 "is th
e following consequence of a deep " }{TEXT -1 7 "[1970] " }{TEXT 752
25 "n-dimensional theorem of " }{TEXT 336 10 "W. Schmidt" }{TEXT 335
187 " ... (When the manuscript of Schmidt's proof first became availab
le, it provided a Diophantine approximation seminar at the University \+
of Illinois with material for an entire semester...)" }}{PARA 258 ""
0 "" {TEXT -1 10 "__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 5 "From " }{TEXT 325 14 "Jean Dieudonn\351" }{TEXT
-1 3 "'s " }{TEXT 337 31 "A PANORAMA OF PURE MATHEMATICS " }{TEXT -1
8 "(1982): " }{TEXT 264 9 "This book" }{TEXT -1 1 "[" }{TEXT 273 2 "'s
" }{TEXT -1 6 "] ... " }{TEXT 274 21 "aim is to provide an " }{TEXT
-1 9 "extremely" }{TEXT 266 1 " " }{TEXT -1 7 "sketchy" }{TEXT 267
168 " survey of a rather large area of modern mathematics, and a guide
to the literature for those who wish to embark on a more serious stud
y of any of the subjects surveyed" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "At the end of each surve
yed topic, Dieudonn\351 has an 'originators' section. I quote from the
end of his Theory of Numbers chapter [I include " }{TEXT 896 4 "only
" }{TEXT -1 71 " names of mathematicians who figure in my talk, howeve
r peripherally]: " }{TEXT 265 85 "The principal ideas in the theory of
numbers are due to the following mathematicians:" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 24 "Algebraic
number theory." }{TEXT -1 143 " L. Euler (1707-1783), J. L. Lagrange
(1736-1813), C. Hermite (1822-1901), D. Hilbert (1862-1943), C. Siege
l (1896-1081) (and nineteen others)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 55 "Diophantine approximatio
ns and transcendental numbers. " }{TEXT -1 418 "C. Hermite (1822-1901)
, A. Thue (1863-1922), C. Siegel (1896-1981), A. Gelfond (1906-1968), \+
T. Schneider, K. Roth, A. Baker, W. Schmidt. [complete list of all tho
se named by Dieudonn\351. Klaus Roth and Alan Baker are Fields Medalli
sts (1958 and 1970). Wolfgang Schmidt must surely be one of the greate
st mathematicians (Andrew Wiles being another) not to have been awarde
d a Fields medal on the foolish age-restriction.]" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 22 "Diophanti
ne geometry. " }{TEXT -1 63 "L. Mordell (1888-1972), C. Siegel (1896-1
981) (and five others)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }{TEXT 271 19 "Arithmetic groups. " }{TEXT -1 64 "C
. Hermite (1822-1901), C. Siegel (1896-1981) (and seven others)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT
272 23 "Analytic number theory." }{TEXT -1 40 " G. Hardy (1877-1947) (
and seven others)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 308 "The following have also contributed substantially to the
se theories: H. Davenport (1907-1969), N. Feldman, E. Landau (1877-193
8), S. Lang, C. Lindemann (1852-1939), Ju. Linnik (1915-1972), J. Liou
ville (1809-1882), K. Mahler (1903-1988), C. Rogers, S. Schanuel, P. T
ur\341n (1910-1976) (and ninety-eight others)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Incidentally, Dieudonn
\351 placed A. Wiles in the later group (remember it was 1982). Where \+
would " }{TEXT 275 3 "his" }{TEXT -1 70 " name be in a new edition? I \+
think we all know the answer to that one." }}}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 20 "Irrational numbers (" }{TEXT 345 7 "briefly" }{TEXT -1 2
") " }{TEXT 528 1 "e" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi;" "6#%#Pi
G" }{TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "A very brief \+
introduction" }}{PARA 0 "" 0 "" {TEXT -1 30 "I suppose everyone knows \+
that " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 67 " is
irrational, and has read the standard textbook proof, with its " }
{TEXT 977 46 "we may suppose without loss of generality that" }{TEXT
-1 1 " " }{XPPEDIT 18 0 "m/n;" "6#*&%\"mG\"\"\"%\"nG!\"\"" }{TEXT -1
1 " " }{TEXT 978 21 "is in reduced form..." }{TEXT -1 5 " . I " }
{TEXT 976 4 "hate" }{TEXT -1 67 " the standard textbook presentation o
f the proof. I don't hate the " }{TEXT 979 5 "proof" }{TEXT -1 20 "; r
ather I hate the " }{TEXT 980 12 "presentation" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 56 " Sadly (in my view) youngsters don't g
et a chance to " }{TEXT 968 5 "think" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }{TEXT 969 6 "before" }
{TEXT -1 205 " being exposed to its brutal irrationality proof that ra
mbles on about \"we may suppose that ... may be expressed in reduced f
orm\" (even G. H. Hardy - an early hero of mine - is guilty of it in h
is classic " }{TEXT 970 25 "A Mathematician's Apology" }{TEXT -1 14 ")
. Youngsters " }{TEXT 972 6 "should" }{TEXT -1 28 " be given an opport
unity to " }{TEXT 971 7 "attempt" }{TEXT -1 93 " to find a 'fraction' \+
- a term that youngsters probably prefer over 'rational number' - that
" }{TEXT 973 8 "possibly" }{TEXT -1 8 " equals " }{XPPEDIT 18 0 "sqrt
(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 131 " (of course they're in for a \+
surprise). Since that is a topic on which I have already written at co
nsiderable length in my 56-page " }{TEXT 974 17 "Number theorising" }
{TEXT -1 1 " " }{TEXT 975 19 "with Talented Youth" }{TEXT -1 90 " (ava
ilable from the Talented Youth corner of my web site) then I will not \+
pursue it here." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 1012 159 "
Briefly, how do I think irrationality (as a subject to be contemplated
) can/should be introduced to youngsters? From my experience, many you
ngsters think that " }{XPPEDIT 269 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }
{TEXT 981 1 " " }{TEXT 982 2 "is" }{TEXT 983 43 " 1.414... (the calcul
ator displayed value):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ev
alf(sqrt(2), 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 984 23 "A thoughtful student - \+
" }{TEXT 985 12 "when pressed" }{TEXT 986 33 " - will tell you that 1.
414213562" }{TEXT -1 1 " " }{TEXT 987 3 "is " }{TEXT 990 3 "not" }
{TEXT 991 1 " " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT
-1 1 " " }{TEXT 988 32 "because, when you square it, you" }{TEXT -1 1
" " }{TEXT 989 126 "get a decimal that ends in a 4, and so it can't be
2. More pressing (that could be motivated by by further computations \+
like:)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalf(sqrt(2), 40)
; # this \"really ends\" in '7'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 992 60 "will lead -
with proof (the crux!) - to the conclusion that " }{XPPEDIT 18 0 "sqr
t(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }{TEXT 993 107 "does not ha
ve a terminating decimal value. That's an elementary, but serious resu
lt. (Aside. What, though, " }{TEXT 1015 2 "is" }{TEXT 1016 38 " the no
n-terminating decimal value of " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG
6#\"\"#" }{TEXT -1 0 "" }{TEXT 1014 212 "; what are its digits? Once, \+
when I was a student in London, I sat with a group of other students i
n the company of C. A. Rogers. Someone asked him which question he wou
ld most like to have answered. He replied: " }{TEXT 1017 55 "to know t
he decimal expansion of the square-root of two" }{TEXT 1018 331 ". We \+
all knew what he meant. Many years later, in June 1986, I travelled to
London to attend the UCL meeting marking C.A.R.'s retirement. Looking
back over his life, Rogers remarked on his good fortune to have had a
good teacher at school, and he mentioned the friendly competition tha
t he had later enjoyed with Wolfgang Schmidt.)" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 1013 36 "Next, in my view (ba
sed on teaching " }{TEXT 994 10 "experience" }{TEXT 1006 236 ") the di
scussion can continue on this line of observation (followed by appropr
iate questioning): numbers with terminating decimal values are simply \+
fractions with a very particular type of denominator (powers of 10), a
nd so saying (e.g)" }{TEXT -1 1 " " }{TEXT 1007 5 "that " }{XPPEDIT
18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }{TEXT 1008 3 "is \+
" }{TEXT 995 3 "not" }{TEXT -1 1 " " }{TEXT 1009 34 "1.414 is simply s
aying that it is " }{TEXT 996 3 "not" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
1414/1000;" "6#*&\"%99\"\"\"\"%+5!\"\"" }{TEXT -1 0 "" }{TEXT 1010 48
", but (the motivated question) could it be (say)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "1414/999;" "6#*&\"%99\"\"\"\"$***!\"\"" }{TEXT 1011 18
"? Well, let's see:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf
(1414/999);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0
"" 0 "" {TEXT -1 30 "So, it isn't. But could it be " }{TEXT 997 14 "so
mething else" }{TEXT -1 41 "? Some other fraction? Could it be (say) \+
" }{XPPEDIT 18 0 "47321/33461;" "6#*&\"&@t%\"\"\"\"&hM$!\"\"" }{TEXT
-1 3 "?: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalf(sqrt(2), \+
10);\nevalf(47321/33461, 10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109
"No. And why not? Resorting to computation we see they aren't equal, a
s they differ on the last decimal point:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 43 "evalf(sqrt(2), 11);\nevalf(47321/33461, 11);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "But could " }{XPPEDIT 18 0 "sqrt(2
);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 10 " be (say) " }{XPPEDIT 18 0 "4478
554083/3166815962;" "6#*&\"+$3a&yW\"\"\"\"+if\"o;$!\"\"" }{TEXT -1 2 "
?:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(sqrt(2), 20);\n
" }{TEXT -1 0 "" }{MPLTEXT 1 0 33 "evalf(4478554083/3166815962, 20);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 15 "Again, no. But " }{TEXT 998 3 "why" }{TEXT -1 145 " not
? And this is the point at which one can start asking for reasons ind
ependent of computation. Of course it's also the time to start asking:
" }{TEXT 1004 5 "where" }{TEXT -1 203 " are those fantastic fractions
coming from? It's the sort of thing - if you don't know about this - \+
you may read about in the Talented Youth corner of my web site, or in
my ICTMT4 Plymouth 1999 talk on " }{TEXT 999 23 "L- and R-approximati
ons" }{TEXT -1 75 ", or in the first year section of my Courses I Teac
h corner of my web site." }}{PARA 0 "" 0 "" {TEXT -1 8 " Why " }
{TEXT 1000 5 "can't" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4478554083/316681
5962" "6#*&\"+$3a&yW\"\"\"\"+if\"o;$!\"\"" }{TEXT -1 4 " be " }
{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 23 "? Suppose i
t was, then " }{XPPEDIT 18 0 "sqrt(2) = 4478554083/3166815962;" "6#/-%
%sqrtG6#\"\"#*&\"+$3a&yW\"\"\"\"+if\"o;$!\"\"" }{TEXT -1 11 ", and the
n?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0
"sqrt(2)^2 = (4478554083/3166815962)^2;" "6#/*$-%%sqrtG6#\"\"#F(*$*&\"
+$3a&yW\"\"\"\"+if\"o;$!\"\"F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "2 = (
4478554083/3166815962)^2;" "6#/\"\"#*$*&\"+$3a&yW\"\"\"\"+if\"o;$!\"\"
F$" }{TEXT -1 10 ", and?\n\n2*" }{XPPEDIT 18 0 "3166815962^2 = 4478554
083^2;" "6#/*$\"+if\"o;$\"\"#*$\"+$3a&yWF&" }{TEXT -1 17 " (and then w
hat?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191
"Impossible since... (e.g.: the LHS ends in '8', the RHS ends in '9'; \+
the LHS is even, the RHS is odd). That's not the end of it, of course.
But it all eventually leads to having a proof that " }{XPPEDIT 18 0 "
sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 4 " is " }{TEXT 1001 10 "irrat
ional" }{TEXT -1 344 ". In short, by following a well-motivated path, \+
one may lead young minds to a basic understanding of the fundamental m
athematical concept of irrationality. If we are interested in teaching
, then this - or something like it - is what we should be doing. Incid
entally, a much faster route into irrationality may be given by asking
questions like:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 ""
{TEXT 1002 2 "Is" }{TEXT -1 1 " " }{XPPEDIT 18 0 "log[10](2) = .3010;
" "6#/-&%$logG6#\"#56#\"\"#$\"%5I!\"%" }{TEXT -1 38 " (as the old log \+
tables used to show)?" }}{PARA 15 "" 0 "" {TEXT 1003 2 "Is" }{TEXT -1
1 " " }{XPPEDIT 18 0 "log[10](2) = .3010299957;" "6#/-&%$logG6#\"#56#
\"\"#$\"+d**H5I!#5" }{TEXT -1 45 " (as ones calculator may show, or, i
n Maple)?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(log[10](
2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 32 "Then, having easily argued that " }{XPPEDIT 18 0
"log[10](2)" "6#-&%$logG6#\"#56#\"\"#" }{TEXT -1 127 " does not have a
terminating decimal value, move on to easily argue that it is, in fac
t, irrational. While numbers like (e.g.) " }{XPPEDIT 18 0 "sqrt(2);" "
6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "log[10](2);" "6
#-&%$logG6#\"#56#\"\"#" }{TEXT -1 110 " are both irrational - the latt
er having the easier irrationality proof - these two numbers are funda
mentally " }{TEXT 1005 15 "quite different" }{TEXT -1 158 ": the forme
r is an algebraic number, while the latter is a transcendental number.
But I am getting ahead of myself. It is time I moved on to the next s
ection." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 ""
{TEXT -1 0 "" }{TEXT 1019 1 "e" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi
;" "6#%#PiG" }{TEXT -1 18 " (irrationality). " }{XPPEDIT 18 0 "Pi <> 2
2/7;" "6#0%#PiG*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 21 "The irrationality of " }{TEXT 1025 1 "e" }{TEXT -1 77 "
was first established by Euler, who concluded it from his derivation \+
of the " }{TEXT 1026 8 "infinite" }{TEXT -1 1 " " }{TEXT 1027 18 "cont
inued fraction" }{TEXT -1 15 " expansion of " }{XPPEDIT 18 0 "(e-1)/2
;" "6#*&,&%\"eG\"\"\"F&!\"\"F&\"\"#F'" }{TEXT -1 169 ". I will not dwe
ll much on this since I cannot presume a familarity with continued fra
ction theory on behalf of a general reader. However I show some brief \+
computations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "with(numthe
ory): # for use of 'cfrac'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
57 "cfrac((exp(1) - 1)/2, 5); # to the 5th 'partial quotient'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 53 "where the pattern from the partial quotient '6' (the " }
{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 59 " of the standard
continued fraction notation) is continued " }{TEXT 1028 12 "ad infini
tum" }{TEXT -1 1 ":" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "a[0]+1/(a[1]+1
/(a[2]+1/(a[3]+1/(a[4]+1/`...`))));" "6#,&&%\"aG6#\"\"!\"\"\"*&F(F(,&&
F%6#F(F(*&F(F(,&&F%6#\"\"#F(*&F(F(,&&F%6#\"\"$F(*&F(F(,&&F%6#\"\"%F(*&
F(F(%$...G!\"\"F(F>F(F>F(F>F(F>F(" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 41 "cfrac ((exp(1) - 1)/2, 50, 'quotients'); " }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 76 "Everyone knows the standard classic irra
tionality proof of (Euler's number) " }{TEXT 1020 1 "e" }{TEXT -1 53 "
, the proof, given by Fourier in 1815, that exploits:" }}{PARA 258 ""
0 "" {TEXT -1 0 "" }{TEXT 1021 1 "e" }{TEXT -1 3 " = " }{XPPEDIT 18 0
"Sum(1/n!,n = 0 .. infinity) = 1+1/1!+1/2!+1/3!+`...`+1/n!+`...`;" "6#
/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!%)infinityG,0F(F
(*&F(F(-F*6#F(F-F(*&F(F(-F*6#\"\"#F-F(*&F(F(-F*6#\"\"$F-F(%$...GF(*&F(
F(-F*6#F,F-F(F>F(" }{TEXT -1 9 " (i)" }}{PARA 0 "" 0 "" {TEXT -1
8 "Suppose " }{XPPEDIT 18 0 "e = a/b;" "6#/%\"eG*&%\"aG\"\"\"%\"bG!\"
\"" }{TEXT -1 35 " for some (positive) whole numbers " }{TEXT 1022 1 "
a" }{TEXT -1 5 " and " }{TEXT 1023 1 "b" }{TEXT -1 37 ". Then multiply
ing throughout (i) by " }{TEXT 1024 1 "n" }{TEXT -1 8 "! gives:" }}
{PARA 258 "" 0 "" {XPPEDIT 18 0 "n!*e = n!+n!/1!+n!/2!+n!/3!+`...`+n!/
n!+n!*R[n];" "6#/*&-%*factorialG6#%\"nG\"\"\"%\"eGF),0-F&6#F(F)*&-F&6#
F(F)-F&6#F)!\"\"F)*&-F&6#F(F)-F&6#\"\"#F3F)*&-F&6#F(F)-F&6#\"\"$F3F)%$
...GF)*&-F&6#F(F)-F&6#F(F3F)*&-F&6#F(F)&%\"RG6#F(F)F)" }{TEXT -1 12 " \+
... (ii)" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 258 "" 0 "
" {XPPEDIT 18 0 "0 < R[n];" "6#2\"\"!&%\"RG6#%\"nG" }{TEXT -1 3 " = "
}{XPPEDIT 18 0 "1/(n+1)!+1/(n+2)!+`...` < 2/(n+1)!;" "6#2,(*&\"\"\"F&-
%*factorialG6#,&%\"nGF&F&F&!\"\"F&*&F&F&-F(6#,&F+F&\"\"#F&F,F&%$...GF&
*&F1F&-F(6#,&F+F&F&F&F," }{TEXT -1 11 ", and thus " }{XPPEDIT 18 0 "n!
*R[n] < 2/(n+1);" "6#2*&-%*factorialG6#%\"nG\"\"\"&%\"RG6#F(F)*&\"\"#F
),&F(F)F)F)!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 59 "Thus (i) is impossible since (ii) - for s
ufficiently large " }{TEXT 1029 1 "n" }{TEXT -1 115 " - leads to an in
teger (the LHS) being equal to (an integer, plus a positive error term
that is less than 1). Thus " }{TEXT 1030 1 "e" }{TEXT -1 15 " is irra
tional." }}{PARA 258 "" 0 "" {TEXT -1 9 "_________" }}{PARA 0 "" 0 ""
{TEXT -1 70 "Any obvious attempt at giving a similar irrationality pro
of for (say) " }{XPPEDIT 18 0 "e^2;" "6#*$%\"eG\"\"#" }{TEXT -1 251 "a
ppears at first sight to be doomed (try it to see what I mean). In his
1949 Princeton book on Transcendental Numbers, C. L. Siegel gave an e
lementary proof - which I had always assumed was his own, since he did
n't credit it to anyone - not only that " }{XPPEDIT 18 0 "e^2;" "6#*$%
\"eG\"\"#" }{TEXT -1 40 " is irrational, but more generally that " }
{TEXT 1031 1 "e" }{TEXT -1 43 " is not a quadratic algebraic number (i
.e. " }{TEXT 1032 1 "e" }{TEXT -1 41 " does not satisfy any quadratic \+
equation " }{XPPEDIT 18 0 "a*e^2+b*e+c = 0;" "6#/,(*&%\"aG\"\"\"*$%\"e
G\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 15 " with integers "
}{TEXT 1033 1 "a" }{TEXT -1 2 ", " }{TEXT 1034 1 "b" }{TEXT -1 5 " and
" }{TEXT 1035 1 "c" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "a <> 0;" "6#
0%\"aG\"\"!" }{TEXT -1 139 "). While preparing my Manchester 1972-73 c
ourse I made a minor improvement on Siegel's proof, by formulating ano
ther elementary proof that " }{XPPEDIT 18 0 "e^2;" "6#*$%\"eG\"\"#" }
{TEXT -1 134 "is irrational. That alternative proof had the advantage \+
that incorporating the idea in Siegel's proof led to an elementary pro
of that " }{XPPEDIT 18 0 "e^4;" "6#*$%\"eG\"\"%" }{TEXT -1 16 " is irr
ational. " }}{PARA 0 "" 0 "" {TEXT -1 248 " Although I gave a 'spli
nter group' talk (attended by Masser and Serre, who clearly had nothin
g better to do) on my proof at the BMC in 1973(?) I didn't bother to w
rite up the proof for possible publication until 2002. I submitted it \+
to the MAA " }{TEXT 1036 7 "Monthly" }{TEXT -1 338 " in that year, and
was just a little disappointed to learn from a scholarly referee that
Siegel's proof had already been given by Liouville in 1840, and that \+
'my' proof was also given by Liouville in the same year (though I beli
eve I give a better explanation that Liouville!). An interested reader
may consult the submitted paper in the " }{TEXT 1037 8 "esquared" }
{TEXT -1 23 " corner of my web site." }}{PARA 0 "" 0 "" {TEXT -1 112 "
Of course all of these elementary results are completely put in th
e shade with Hermite's (1873) result that " }{TEXT 1038 1 "e" }{TEXT
-1 28 " is a transcendental number." }}{PARA 258 "" 0 "" {TEXT -1 10 "
__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 21 "The irrationality of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT
-1 105 " was first demonstrated by Lambert in 1766. As with Euler, con
tinued fractions were central to his proof." }}{PARA 0 "" 0 "" {TEXT
-1 41 " There are many different proofs that " }{XPPEDIT 18 0 "Pi;
" "6#%#PiG" }{TEXT -1 140 " is irrational, but it should be said that \+
none of them are elementary. Here my sole aim is to show you a very be
autiful way of seeing that " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1
22 " isn't the oft-quoted " }{XPPEDIT 18 0 "22/7;" "6#*&\"#A\"\"\"\"\"
(!\"\"" }{TEXT -1 378 ". I believe this could be understood, and how n
ice it would be to have it appreciated, by competent school pupils. I \+
only came to know of this way in August 1996 while browsing in a books
hop in Blackwell's of Oxford: there I came upon a lovely integral, in \+
a paper by van der Poorten and Bombieri, which I had never seen before
: an integral with positive integrand, with value (" }{XPPEDIT 18 0 "2
2/7-Pi;" "6#,&*&\"#A\"\"\"\"\"(!\"\"F&%#PiGF(" }{TEXT -1 200 "). What \+
I saw fairly set my heart thumping, and I read no further as I wanted \+
to have an opportunity to play. This is what I saw in their paper, and
I hardly need comment on the obvious implications: " }}{PARA 258 ""
0 "" {XPPEDIT 18 0 "Int(x^4*(1-x)^4/(1+x^2),x = 0 .. 1) = 22/7-Pi;" "6
#/-%$IntG6$*(%\"xG\"\"%,&\"\"\"F+F(!\"\"F),&F+F+*$F(\"\"#F+F,/F(;\"\"!
F+,&*&\"#AF+\"\"(F,F+%#PiGF," }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "It's an easy school exercise t
o evaluate that integral, and I quickly show that Maple can cope with \+
it, and more:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(x^4*(1 \+
- x)^4/(1 + x^2), x = 0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 39 "int(x^4*(1 - x)^4/(1 + x^2), x = 0..1);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 17 "evalf(22/7 - Pi);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 74 "plot(x^4*(1 - x)^4/(1 + x^2), x = 0..1); \n# Notice t
he scale on the y-axis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "But th
ere is so much more to be investigated, discovered, and " }{TEXT 1041
6 "proved" }{TEXT -1 152 ". Can one explain this, can one explain that
...? I did a lot of related work in August 1996, but have never done a
nything about it. I drop some passing " }{TEXT 1039 5 "hints" }{TEXT
-1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(x^40*(1 - x)^
40/(1 + x^2), x = 0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16
"ifactor(262144);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "ifacto
r(216850257105757801880233554675); \n# the denominator of the approxim
ating denominator" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "I need hardly point out that all t
he odd primes between 3 and 79 occur there, except for 19." }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(x^68*(1 - x)^68/(1 + x^2), x = \+
0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ifactor(429496729
6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "ifactor(98251148449
89333870090599983047666848673348322555875);\n# the denominator of the \+
approximating denominator" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0
"" }}}{PARA 0 "" 0 "" {TEXT -1 75 "One will notice that the primes 13 \+
and 107 are 'missing' between 3 and 131." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "And here I have varied the powers \+
in the integrand:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(x^5
2*(1 - x)^76/(1 + x^2), x = 0..1);#" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 21 "ifactor(68719476736);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 112 "ifactor(28740005361134236353744294947572823137351223
9326125);\n# the denominator of the approximating denominator" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 35 "An obvious question to ask is: can " }{TEXT 1040 18 "some such \+
integral" }{TEXT -1 30 " be used to give a proof that " }{XPPEDIT 18
0 "Pi;" "6#%#PiG" }{TEXT -1 15 " is irrational?" }}}{SECT 1 {PARA 4 "
" 0 "" {TEXT 1071 65 "The meaning of 'linearly independent over the in
tegers/rationals'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 129 "Later, if you read the Roth section, and
then the Schmidt and Baker sections, there is a standard expression y
ou will encounter: " }{TEXT 1042 46 "'... linearly independent over th
e rationals' " }{TEXT -1 36 " (which is equivalent in meaning to " }
{TEXT 1055 42 "' linearly independent over the integers' " }{TEXT -1
89 "). I would like to explain what that expression means, because wit
hout understanding the " }{TEXT 1056 7 "meaning" }{TEXT -1 55 " of thi
s expression one cannot begin to appreciate the " }{TEXT 1043 12 "sign
ificance" }{TEXT -1 58 " of the use of it in the Roth, Schmidt and Bak
er sections." }}{PARA 0 "" 0 "" {TEXT -1 33 " Choose any irrational
number " }{TEXT 1044 1 "A" }{TEXT -1 58 ". Then, infinitely many othe
r numbers are irrational as a " }{TEXT 1072 11 "consequence" }{TEXT
-1 15 ". For example 2" }{TEXT 1045 1 "A" }{TEXT -1 3 ", 3" }{TEXT
1046 1 "A" }{TEXT -1 4 ", -4" }{TEXT 1047 1 "A" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "3*A/7;" "6#*(\"\"$\"\"\"%\"AGF%\"\"(!\"\"" }{TEXT -1 2
", " }{XPPEDIT 18 0 "-4*A/11;" "6#,$*(\"\"%\"\"\"%\"AGF&\"#6!\"\"F)" }
{TEXT -1 10 ", ... are " }{TEXT 1050 14 "all irrational" }{TEXT -1 49
". What numbers are being suggested there? Simply " }{TEXT 1049 27 "no
n-zero rational multiples" }{TEXT -1 4 " of " }{TEXT 1048 1 "A" }
{TEXT -1 14 ". Thus, e.g., " }{XPPEDIT 18 0 "3*A/7" "6#*(\"\"$\"\"\"%
\"AGF%\"\"(!\"\"" }{TEXT -1 54 " is (trivially) irrational, for if not
, we would have " }{XPPEDIT 18 0 "3*A/7 = m/n;" "6#/*(\"\"$\"\"\"%\"AG
F&\"\"(!\"\"*&%\"mGF&%\"nGF)" }{TEXT -1 28 " for some integers m and n
(" }{XPPEDIT 18 0 "n <> 0;" "6#0%\"nG\"\"!" }{TEXT -1 10 "), giving \+
" }{XPPEDIT 18 0 "A = 7*m/(3*n);" "6#/%\"AG*(\"\"(\"\"\"%\"mGF'*&\"\"$
F'%\"nGF'!\"\"" }{TEXT -1 23 ", is rational, whereas " }{TEXT 1051 1 "
A" }{TEXT -1 15 " is irrational." }}{PARA 0 "" 0 "" {TEXT -1 77 " N
ow - to make a point - I choose two irrational numbers, two old friend
s " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " and "
}{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 28 ", and ask:
could it be that " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }
{TEXT -1 20 " is irrational as a " }{TEXT 1052 6 "simple" }{TEXT -1
13 " consequence " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }
{TEXT -1 44 "'s irrationality. Meaning? Could it be that " }{XPPEDIT
18 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT -1 6 " is a " }{TEXT 1053 8
"rational" }{TEXT -1 13 " multiple of " }{XPPEDIT 18 0 "sqrt(2);" "6#-
%%sqrtG6#\"\"#" }{TEXT -1 52 "? Well it isn't (and a novice might like
to wonder: " }{TEXT 1054 8 "why not?" }{TEXT -1 24 "). Neither of cou
rse is " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 24 " \+
a rational multiple of " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 " Whereas if I had
chosen as my two irrationals (e.g.) " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%
sqrtG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sqrt(50/9);" "6#-%%s
qrtG6#*&\"#]\"\"\"\"\"*!\"\"" }{TEXT -1 37 " , then here each is irrat
ional as a " }{TEXT 1057 6 "simple" }{TEXT -1 27 " consequence of the \+
other: " }{XPPEDIT 18 0 "sqrt(50/9);" "6#-%%sqrtG6#*&\"#]\"\"\"\"\"*!
\"\"" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "5/3;" "6#*&\"\"&\"\"\"\"\"$!
\"\"" }{TEXT -1 1 "." }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }
{TEXT -1 14 ", and equally " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"
\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "3/5;" "6#*&\"\"$\"\"\"\"\"&!\"
\"" }{TEXT -1 1 "." }{XPPEDIT 18 0 "sqrt(50/9);" "6#-%%sqrtG6#*&\"#]\"
\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 98 " The (standard) expression that is use
d to summarise what I've just pointed out is to say that:" }}{PARA 15
"" 0 "" {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " and
" }{XPPEDIT 18 0 "sqrt(50/9)" "6#-%%sqrtG6#*&\"#]\"\"\"\"\"*!\"\"" }
{TEXT -1 5 " are " }{TEXT 1058 18 "linearly dependent" }{TEXT -1 1 " \+
" }{TEXT 1059 18 "over the rationals" }{TEXT -1 14 " (or integers)" }}
{PARA 15 "" 0 "" {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT
-1 5 " and " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1
5 " are " }{TEXT 1060 9 "linearly " }{TEXT 1061 2 "in" }{TEXT 1062 28
"dependent over the rationals" }{TEXT -1 14 " (or integers)" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Formal definiti
on. Let " }{XPPEDIT 18 0 "alpha[1],alpha[2],`...`,alpha[n];" "6&&%&alp
haG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 4 " be " }{TEXT 1063
1 "n" }{TEXT -1 34 " (real or complex) numbers (above " }{TEXT 1064 1
"n" }{TEXT -1 12 " = 2), then " }{XPPEDIT 18 0 "alpha[1], alpha[2], `.
..`, alpha[n]" "6&&%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT
-1 16 " are said to be " }{TEXT 1066 37 "linearly dependent over the r
ationals" }{TEXT -1 38 " (equivalently integers) if there are " }
{TEXT 1067 8 "rational" }{TEXT -1 9 " numbers " }{XPPEDIT 18 0 "r[1],r
[2],`...`,r[n];" "6&&%\"rG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT
-1 3 " - " }{TEXT 1065 12 "not all zero" }{TEXT -1 13 " - such that "
}{XPPEDIT 18 0 "r[1]*alpha[1]+r[2]*alpha[2]+`...`+r[n]*alpha[n] = 0;"
"6#/,**&&%\"rG6#\"\"\"F)&%&alphaG6#F)F)F)*&&F'6#\"\"#F)&F+6#F0F)F)%$..
.GF)*&&F'6#%\"nGF)&F+6#F7F)F)\"\"!" }{TEXT -1 5 ". If " }{XPPEDIT 18
0 "alpha[1], alpha[2], `...`, alpha[n]" "6&&%&alphaG6#\"\"\"&F$6#\"\"#
%$...G&F$6#%\"nG" }{TEXT -1 5 " are " }{TEXT 1068 3 "not" }{TEXT -1
105 " linearly dependent over the rationals then they are said to be l
inearly independent over the rationals. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 1. " }{XPPEDIT 18 0 "alpha
[1] = sqrt(2);" "6#/&%&alphaG6#\"\"\"-%%sqrtG6#\"\"#" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "alpha[2] = sqrt(50/9);" "6#/&%&alphaG6#\"\"#-%%sqr
tG6#*&\"#]\"\"\"\"\"*!\"\"" }{TEXT -1 23 " are linearly dependent" }
{TEXT 1069 1 " " }{TEXT -1 33 "over the rationals since we have " }
{XPPEDIT 18 0 "alpha[1] = 3*alpha[2]/5;" "6#/&%&alphaG6#\"\"\"*(\"\"$F
'&F%6#\"\"#F'\"\"&!\"\"" }{TEXT -1 11 ", and thus " }{XPPEDIT 18 0 "al
pha[1]-3*alpha[2]/5 = 0;" "6#/,&&%&alphaG6#\"\"\"F(*(\"\"$F(&F&6#\"\"#
F(\"\"&!\"\"F/\"\"!" }{TEXT -1 5 " (so " }{XPPEDIT 18 0 "r[1];" "6#&%
\"rG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r[2];" "6#&%\"rG6#\"
\"#" }{TEXT -1 41 " may be chosen to be the rationals 1 and " }
{XPPEDIT 18 0 "-3/5;" "6#,$*&\"\"$\"\"\"\"\"&!\"\"F(" }{TEXT -1 60 ", \+
but equally they could be chosen to be the integers 5 and " }{XPPEDIT
18 0 "-3;" "6#,$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Example 2 (exercise). " }
{XPPEDIT 18 0 "alpha[1] = log[10](4);" "6#/&%&alphaG6#\"\"\"-&%$logG6#
\"#56#\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha[2] = log[10](8)
;" "6#/&%&alphaG6#\"\"#-&%$logG6#\"#56#\"\")" }{TEXT -1 23 " are linea
rly dependent" }{TEXT 1070 1 " " }{TEXT -1 86 "over the rationals (and
notice it doesn't matter what is the base of the logarithms). " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 3
. " }{XPPEDIT 18 0 "alpha[1] = log[10](5);" "6#/&%&alphaG6#\"\"\"-&%$l
ogG6#\"#56#\"\"&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha[2] = log[1
0](7);" "6#/&%&alphaG6#\"\"#-&%$logG6#\"#56#\"\"(" }{TEXT -1 5 " are \+
" }{TEXT 1073 20 "linearly independent" }{TEXT -1 1 " " }{TEXT 1074
18 "over the rationals" }{TEXT -1 19 ". For if not, then " }{XPPEDIT
18 0 "r[1]*log[10](5)+r[2]*log[10](7) = 0;" "6#/,&*&&%\"rG6#\"\"\"F)-&
%$logG6#\"#56#\"\"&F)F)*&&F'6#\"\"#F)-&F,6#F.6#\"\"(F)F)\"\"!" }{TEXT
-1 20 " for some rationals " }{XPPEDIT 18 0 "r[1],r[2];" "6$&%\"rG6#\"
\"\"&F$6#\"\"#" }{TEXT -1 26 ", not both zero. But then " }{XPPEDIT
18 0 "log[10](5^r[1])+log[10](7^r[2]) = 0;" "6#/,&-&%$logG6#\"#56#)\"
\"&&%\"rG6#\"\"\"F0-&F'6#F)6#)\"\"(&F.6#\"\"#F0\"\"!" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "log[10](5^r[1]*7^r[2]) = 0;" "6#/-&%$logG6#\"#56#*&)
\"\"&&%\"rG6#\"\"\"F0)\"\"(&F.6#\"\"#F0\"\"!" }{TEXT -1 9 ", giving "
}{XPPEDIT 18 0 "5^r[1]*7^r[2] = 1;" "6#/*&)\"\"&&%\"rG6#\"\"\"F*)\"\"(
&F(6#\"\"#F*F*" }{TEXT -1 40 ", which is impossible (why?) except for \+
" }{XPPEDIT 18 0 "r[1],r[2];" "6$&%\"rG6#\"\"\"&F$6#\"\"#" }{TEXT -1
1 " " }{TEXT 1075 4 "both" }{TEXT -1 12 " being zero." }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Example 4 (exercise). \+
" }{XPPEDIT 18 0 "alpha[1] = log[10](45);" "6#/&%&alphaG6#\"\"\"-&%$lo
gG6#\"#56#\"#X" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "alpha[2] = log[10](27
);" "6#/&%&alphaG6#\"\"#-&%$logG6#\"#56#\"#F" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "alpha[3] = log[10](25);" "6#/&%&alphaG6#\"\"$-&%$logG6#
\"#56#\"#D" }{TEXT -1 4 "are " }{TEXT 1076 18 "linearly dependent" }
{TEXT -1 1 " " }{TEXT 1077 18 "over the rationals" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "I now ma
ke one last elementary point. First, return to the the earlier 'Choose
any irrational number " }{TEXT 1079 1 "A" }{TEXT -1 66 "...', and not
e that not only is any non-zero rational multiple of " }{TEXT 1080 1 "
A" }{TEXT -1 28 " irrational, but so also is " }{TEXT 1081 15 "any suc
h number" }{TEXT -1 78 " plus any rational number (positive, negative,
or (trivially) 0). Thus, e.g., " }{XPPEDIT 18 0 "3*sqrt(2)/7-13/11;"
"6#,&*(\"\"$\"\"\"-%%sqrtG6#\"\"#F&\"\"(!\"\"F&*&\"#8F&\"#6F,F," }
{TEXT -1 21 " is irrational (why?)" }}{PARA 0 "" 0 "" {TEXT -1 40 " \+
Now recall that the two irrationals " }{XPPEDIT 18 0 "sqrt(2);" "6#-%
%sqrtG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqr
tG6#\"\"$" }{TEXT -1 216 " are essentially different, in that neither'
s irrationality is a simple consequence of the other because they are \+
(the clarifying) 'linearly independent over the rationals'. But note t
hat while (the similar looking) " }{XPPEDIT 18 0 "alpha[1] = 3/4+2*sqr
t(2)/5;" "6#/&%&alphaG6#\"\"\",&*&\"\"$F'\"\"%!\"\"F'*(\"\"#F'-%%sqrtG
6#F.F'\"\"&F,F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha[2] = 6/7-9*
sqrt(2)/11;" "6#/&%&alphaG6#\"\"#,&*&\"\"'\"\"\"\"\"(!\"\"F+*(\"\"*F+-
%%sqrtG6#F'F+\"#6F-F-" }{TEXT -1 121 " are both irrational (why?), the
y are linearly independent over the rationals (why?), and so neither i
s irrational as an " }{TEXT 1216 9 "immediate" }{TEXT -1 150 " consequ
ence of the other's irrationality. However their respective irrational
ities are intimately linked to each other in that although neither is \+
a " }{TEXT 1078 6 "direct" }{TEXT -1 198 " rational multiple of the ot
her, each however is obtainable from the other by the process I've jus
t outlined: multiply by some non-zero rational, and add some other rat
ional to that. In other words:" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "alph
a[2] = a[1]*alpha[1]+a[2];" "6#/&%&alphaG6#\"\"#,&*&&%\"aG6#\"\"\"F-&F
%6#F-F-F-&F+6#F'F-" }{TEXT -1 27 " for some rational numbers " }
{XPPEDIT 18 0 "a[1],a[2];" "6$&%\"aG6#\"\"\"&F$6#\"\"#" }{TEXT -1 17 "
, or equivalently" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "alpha[1] = b[1]*a
lpha[2]+b[2];" "6#/&%&alphaG6#\"\"\",&*&&%\"bG6#F'F'&F%6#\"\"#F'F'&F+6
#F/F'" }{TEXT -1 27 " for some rational numbers " }{XPPEDIT 18 0 "b[1]
,b[2];" "6$&%\"bG6#\"\"\"&F$6#\"\"#" }{TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 61 "These may be succinctly sumarised by noting that what we \+
are " }{TEXT 1082 13 "really saying" }{TEXT -1 32 " here is that the t
hree numbers " }{XPPEDIT 18 0 "1,alpha[1],alpha[2];" "6%\"\"\"&%&alpha
G6#F#&F%6#\"\"#" }{TEXT -1 79 " are linearly dependent over the ration
als; that is there are rational numbers " }{XPPEDIT 18 0 "r[0],r[1],r[
2];" "6%&%\"rG6#\"\"!&F$6#\"\"\"&F$6#\"\"#" }{TEXT -1 28 " - not all z
ero - such that " }{XPPEDIT 18 0 "r[0]+r[1]*alpha[1]+r[2]*alpha[2] = 0
;" "6#/,(&%\"rG6#\"\"!\"\"\"*&&F&6#F)F)&%&alphaG6#F)F)F)*&&F&6#\"\"#F)
&F.6#F3F)F)F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 35 "I would hope that when you see the " }
{TEXT 1084 15 "proper contexts" }{TEXT -1 139 " - the relevant parts o
f the Roth, Schmidt and Baker sections - these definitions will become
more than mere definitions, and make perfect " }{TEXT 1217 5 "sense"
}{TEXT -1 1 "." }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 55 "Algebraic and transcendental numbers. Euler's intu
ition" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Definitions of algebraic
and transcendental numbers" }}{PARA 0 "" 0 "" {TEXT 597 10 "Motivatio
n" }{TEXT -1 31 " of the definition of the term " }{TEXT 529 16 "algeb
raic number" }{TEXT -1 14 ". The numbers " }{XPPEDIT 18 0 "7/5;" "6#*&
\"\"(\"\"\"\"\"&!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "2^(1/3)+1/5;"
"6#,&)\"\"#*&\"\"\"F'\"\"$!\"\"F'*&F'F'\"\"&F)F'" }{TEXT -1 6 ", and \+
" }{XPPEDIT 18 0 "1/2+sqrt(1-sqrt(2))/2;" "6#,&*&\"\"\"F%\"\"#!\"\"F%*
&-%%sqrtG6#,&F%F%-F*6#F&F'F%F&F'F%" }{TEXT -1 37 " are, in one sense, \+
quite different: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 ""
{TEXT -1 30 "the first is a rational number" }}{PARA 15 "" 0 "" {TEXT
-1 44 "the second is a real irrational number, and " }}{PARA 15 "" 0 "
" {TEXT -1 74 "the third is (clearly) a complex number (with non-zero \+
'imaginary' part) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eval
c(1/2 + sqrt(1 - sqrt(2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
31 "I^2; # in Maple 'I' is sqrt(-1)" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 92 "But in another sense they are not so different: all three
are solutions of equations of the " }{TEXT 571 9 "same kind" }{TEXT
-1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {XPPEDIT
18 0 "7/5;" "6#*&\"\"(\"\"\"\"\"&!\"\"" }{TEXT -1 31 " is a solution o
f the equation " }{XPPEDIT 18 0 "a*x+b = 0;" "6#/,&*&%\"aG\"\"\"%\"xGF
'F'%\"bGF'\"\"!" }{TEXT -1 8 ", where " }{TEXT 530 1 "a" }{TEXT -1 5 "
and " }{TEXT 531 1 "b" }{TEXT -1 15 " are integers: " }{XPPEDIT 18 0
"(a, b) = (5, -7);" "6#/6$%\"aG%\"bG6$\"\"&,$\"\"(!\"\"" }{TEXT -1 0 "
" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "2^(1/3)-1/5;" "6#,&)\"\"#*&\"\"\"F
'\"\"$!\"\"F'*&F'F'\"\"&F)F)" }{TEXT -1 31 " is a solution of the equa
tion " }{XPPEDIT 18 0 "a*x^3+b*x^2+c*x+d = 0;" "6#/,**&%\"aG\"\"\"*$%
\"xG\"\"$F'F'*&%\"bGF'*$F)\"\"#F'F'*&%\"cGF'F)F'F'%\"dGF'\"\"!" }
{TEXT -1 8 ", where " }{TEXT 532 1 "a" }{TEXT -1 2 ", " }{TEXT 533 1 "
b" }{TEXT -1 2 ", " }{TEXT 570 1 "c" }{TEXT -1 5 " and " }{TEXT 534 1
"d" }{TEXT -1 15 " are integers: " }{XPPEDIT 18 0 "(a, b, c, d) = (125
, 75, 15, -249);" "6#/6&%\"aG%\"bG%\"cG%\"dG6&\"$D\"\"#v\"#:,$\"$\\#!
\"\"" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2+sqrt(1-sqr
t(2))/2;" "6#,&*&\"\"\"F%\"\"#!\"\"F%*&-%%sqrtG6#,&F%F%-F*6#F&F'F%F&F'
F%" }{TEXT -1 31 " is a solution of the equation " }{XPPEDIT 18 0 "a*x
^4+b*x^3+c*x^2+d*x+e = 0;" "6#/,,*&%\"aG\"\"\"*$%\"xG\"\"%F'F'*&%\"bGF
'*$F)\"\"$F'F'*&%\"cGF'*$F)\"\"#F'F'*&%\"dGF'F)F'F'%\"eGF'\"\"!" }
{TEXT -1 8 ", where " }{TEXT 572 1 "a" }{TEXT -1 2 ", " }{TEXT 573 4 "
b, c" }{TEXT -1 2 ", " }{TEXT 574 1 "d" }{TEXT -1 5 " and " }{TEXT
575 1 "e" }{TEXT -1 15 " are integers: " }{XPPEDIT 18 0 "(a, b, c, d, \+
e) = (8, -16, 8, 0, -1);" "6#/6'%\"aG%\"bG%\"cG%\"dG%\"eG6'\"\"),$\"#;
!\"\"F+\"\"!,$\"\"\"F." }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 38 "solve(8*x^4 - 16*x^3 + 8*x^2 - 1 = 0);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalc(1/2 + sqrt(1-sqrt(2))/2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 284 10 "DEFINITION" }
{TEXT -1 27 ". A real or complex number " }{XPPEDIT 18 0 "alpha;" "6#%
&alphaG" }{TEXT -1 15 " is said to be " }{TEXT 283 9 "algebraic" }
{TEXT 541 1 " " }{TEXT -1 11 "('over the " }{TEXT 567 8 "integers" }
{TEXT -1 6 "') if " }{XPPEDIT 18 0 "x = alpha;" "6#/%\"xG%&alphaG" }
{TEXT -1 18 " is a solution of " }{TEXT 535 4 "some" }{TEXT -1 21 " po
lynomial equation " }{XPPEDIT 18 0 "a[0]*x^n+a[1]*x^(n-1)+`...`+a[n] =
0;" "6#/,**&&%\"aG6#\"\"!\"\"\")%\"xG%\"nGF*F**&&F'6#F*F*)F,,&F-F*F*!
\"\"F*F*%$...GF*&F'6#F-F*F)" }{TEXT -1 12 ", where the " }{XPPEDIT 18
0 "a[0],a[1],`...`,a[n];" "6&&%\"aG6#\"\"!&F$6#\"\"\"%$...G&F$6#%\"nG
" }{TEXT -1 15 " are integers, " }{XPPEDIT 18 0 "a[0] <> 0;" "6#0&%\"a
G6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }{TEXT 561 7 "REMARK " }{TEXT -1 8 "on that " }
{TEXT 566 12 "DEFINITION. " }{TEXT -1 88 " You can sometimes see an al
gebraic number defined as follows: a real or complex number " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 15 " is said to be " }
{TEXT 562 9 "algebraic" }{TEXT 564 1 " " }{TEXT -1 11 "('over the " }
{TEXT 565 9 "rationals" }{TEXT -1 6 "') if " }{XPPEDIT 18 0 "x = alpha
;" "6#/%\"xG%&alphaG" }{TEXT -1 18 " is a solution of " }{TEXT 563 4 "
some" }{TEXT -1 21 " polynomial equation " }{XPPEDIT 18 0 "a[0]*x^n+a[
1]*x^(n-1)+`...`+a[n] = 0;" "6#/,**&&%\"aG6#\"\"!\"\"\")%\"xG%\"nGF*F*
*&&F'6#F*F*)F,,&F-F*F*!\"\"F*F*%$...GF*&F'6#F-F*F)" }{TEXT -1 12 ", wh
ere the " }{XPPEDIT 18 0 "a[0],a[1],`...`,a[n];" "6&&%\"aG6#\"\"!&F$6#
\"\"\"%$...G&F$6#%\"nG" }{TEXT -1 16 " are rationals, " }{XPPEDIT 18
0 "a[0] <> 0;" "6#0&%\"aG6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 122 " One should appreciate that it doesn't matter which o
f these definitions is adopted as the basis for defining the term " }
{TEXT 568 9 "algebraic" }{TEXT -1 177 ", for - trivially - a number th
at is algebraic according to one of these definitions is automatically
algebraic according to the other definition, and vice versa. (For exa
mple, " }{TEXT 569 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "alpha = 8/3
-2*sqrt(21)/3;" "6#/%&alphaG,&*&\"\")\"\"\"\"\"$!\"\"F(*(\"\"#F(-%%sqr
tG6#\"#@F(F)F*F*" }{TEXT -1 32 " is a solution of the equation " }
{XPPEDIT 18 0 "3*x^2/4-4*x-5/3 = 0;" "6#/,(*(\"\"$\"\"\"*$%\"xG\"\"#F'
\"\"%!\"\"F'*&F+F'F)F'F,*&\"\"&F'F&F,F,\"\"!" }{TEXT -1 9 ", and so "
}{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 72 " is algebraic accor
ding to the latter definition. But clearly that same " }{XPPEDIT 18 0
"alpha;" "6#%&alphaG" }{TEXT -1 31 " is a solution of the equation " }
{XPPEDIT 18 0 "9*x^2-48*x-20 = 0;" "6#/,(*&\"\"*\"\"\"*$%\"xG\"\"#F'F'
*&\"#[F'F)F'!\"\"\"#?F-\"\"!" }{TEXT -1 97 " (the earlier one multipli
ed by the product of the denominators of the rational coefficients) : \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(3/4*x^2 - 4*x - 5
/3 = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(9*x^2 - 4
8*x - 20 = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT
537 10 "DEFINITION" }{TEXT -1 22 ". An algebraic number " }{XPPEDIT
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 18 " is said to be of " }{TEXT
540 6 "degree" }{TEXT -1 1 " " }{TEXT 539 1 "n" }{TEXT -1 4 " if " }
{XPPEDIT 18 0 "x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1 18 " is a solu
tion of " }{TEXT 538 4 "some" }{TEXT -1 21 " polynomial equation " }
{XPPEDIT 18 0 "a[0]*x^n+a[1]*x^(n-1)+`...`+a[n] = 0;" "6#/,**&&%\"aG6#
\"\"!\"\"\")%\"xG%\"nGF*F**&&F'6#F*F*)F,,&F-F*F*!\"\"F*F*%$...GF*&F'6#
F-F*F)" }{TEXT -1 12 ", where the " }{XPPEDIT 18 0 "a[0],a[1],`...`,a[
n];" "6&&%\"aG6#\"\"!&F$6#\"\"\"%$...G&F$6#%\"nG" }{TEXT -1 15 " are i
ntegers (" }{XPPEDIT 18 0 "a[0] <> 0;" "6#0&%\"aG6#\"\"!F'" }{TEXT -1
7 "), but " }{XPPEDIT 18 0 "x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1
106 " is not a solution of any polynomial equation (with integer coeff
icients, not all zero) of smaller degree." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 1. " }{XPPEDIT 18 0 "7/5;
" "6#*&\"\"(\"\"\"\"\"&!\"\"" }{TEXT -1 44 " is clearly an algebraic n
umber of degree 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 11 "Example 2. " }{XPPEDIT 18 0 "(sqrt(7)+3)/5;" "6#*&,&-%%sq
rtG6#\"\"(\"\"\"\"\"$F)F)\"\"&!\"\"" }{TEXT -1 37 " is clearly an alge
braic number (set " }{XPPEDIT 18 0 "alpha = (sqrt(7)+3)/5;" "6#/%&alph
aG*&,&-%%sqrtG6#\"\"(\"\"\"\"\"$F+F+\"\"&!\"\"" }{TEXT -1 7 ", then "
}{XPPEDIT 18 0 "5*alpha-3 = sqrt(7);" "6#/,&*&\"\"&\"\"\"%&alphaGF'F'
\"\"$!\"\"-%%sqrtG6#\"\"(" }{TEXT -1 10 ", and thus" }}{PARA 0 "" 0 "
" {XPPEDIT 18 0 "(5*alpha-3)^2 = sqrt(7)^2;" "6#/*$,&*&\"\"&\"\"\"%&al
phaGF(F(\"\"$!\"\"\"\"#*$-%%sqrtG6#\"\"(F," }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "25*alpha^2-30*alpha+9 = 7;" "6#/,(*&\"#D\"\"\"*$%&alpha
G\"\"#F'F'*&\"#IF'F)F'!\"\"\"\"*F'\"\"(" }{TEXT -1 9 ", and so " }
{XPPEDIT 18 0 "x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1 31 " is a solu
tion of the equation " }{XPPEDIT 18 0 "25*x^2-30*x+2 = 0;" "6#/,(*&\"#
D\"\"\"*$%\"xG\"\"#F'F'*&\"#IF'F)F'!\"\"F*F'\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 6 "Also, " }{XPPEDIT 18 0 "(sqrt(7)+3)/5" "6#
*&,&-%%sqrtG6#\"\"(\"\"\"\"\"$F)F)\"\"&!\"\"" }{TEXT -1 75 ", being ir
rational, cannot be of the first degree, and thus is of degree 2." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Example 3
. " }{XPPEDIT 18 0 "sqrt(2)+sqrt(3);" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#
\"\"$F(" }{TEXT -1 123 " is clearly an algebraic number (but do not ju
mp to the hasty conclusion that it is of degree 2 on the flimsy ground
s that " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 52 " are \+
both algebraic, and of the second degree): set " }{XPPEDIT 18 0 "alpha
= sqrt(2)+sqrt(3);" "6#/%&alphaG,&-%%sqrtG6#\"\"#\"\"\"-F'6#\"\"$F*"
}{TEXT -1 7 ", then " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "alpha^2 = (sqrt
(2)+sqrt(3))^2;" "6#/*$%&alphaG\"\"#*$,&-%%sqrtG6#F&\"\"\"-F*6#\"\"$F,
F&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "5+2*sqrt(6);" "6#,&\"\"&\"\"\"*&
\"\"#F%-%%sqrtG6#\"\"'F%F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(alpha^2-
5)^2 = (2*sqrt(6))^2;" "6#/*$,&*$%&alphaG\"\"#\"\"\"\"\"&!\"\"F(*$*&F(
F)-%%sqrtG6#\"\"'F)F(" }{TEXT -1 13 "= 24, giving " }{XPPEDIT 18 0 "al
pha^4-10*alpha^2+25 = 24;" "6#/,(*$%&alphaG\"\"%\"\"\"*&\"#5F(*$F&\"\"
#F(!\"\"\"#DF(\"#C" }{TEXT -1 13 ", and finally" }}{PARA 0 "" 0 ""
{XPPEDIT 18 0 "alpha^4-10*alpha^2+1 = 0;" "6#/,(*$%&alphaG\"\"%\"\"\"*
&\"#5F(*$F&\"\"#F(!\"\"F(F(\"\"!" }{TEXT -1 7 ". Thus " }{XPPEDIT 18
0 "x = sqrt(2)+sqrt(3);" "6#/%\"xG,&-%%sqrtG6#\"\"#\"\"\"-F'6#\"\"$F*
" }{TEXT -1 45 " is a solution of the fourth degree equation " }
{XPPEDIT 18 0 "x^4-10*x^2+1 = 0;" "6#/,(*$%\"xG\"\"%\"\"\"*&\"#5F(*$F&
\"\"#F(!\"\"F(F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 24 "However, it is not that " }{XPPEDIT
18 0 "sqrt(2)+sqrt(3);" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(" }
{TEXT -1 99 " is of degree 4 simply because of the fact that the equat
ion is of degree 4. After all, if one set " }{XPPEDIT 18 0 "alpha = sq
rt(5);" "6#/%&alphaG-%%sqrtG6#\"\"&" }{TEXT -1 7 ", then " }{XPPEDIT
18 0 "alpha^4 = sqrt(5)^4;" "6#/*$%&alphaG\"\"%*$-%%sqrtG6#\"\"&F&" }
{TEXT -1 14 " = 25, giving " }{XPPEDIT 18 0 "alpha^4-25 = 0;" "6#/,&*$
%&alphaG\"\"%\"\"\"\"#D!\"\"\"\"!" }{TEXT -1 9 ", and so " }{XPPEDIT
18 0 "x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1 45 " is a solution of t
he fourth degree equation " }{XPPEDIT 18 0 "x^4-25 = 0;" "6#/,&*$%\"xG
\"\"%\"\"\"\"#D!\"\"\"\"!" }{TEXT -1 22 ". But, the polynomial " }
{XPPEDIT 18 0 "x^4-25;" "6#,&*$%\"xG\"\"%\"\"\"\"#D!\"\"" }{TEXT -1
99 " happens to factor as the product of two polynomials with integer \+
coefficients, of smaller degree: " }{XPPEDIT 18 0 "x^4-25 = (x^2-5)*(x
^2+5);" "6#/,&*$%\"xG\"\"%\"\"\"\"#D!\"\"*&,&*$F&\"\"#F(\"\"&F*F(,&*$F
&F.F(F/F(F(" }{TEXT -1 15 ", and it's the " }{XPPEDIT 18 0 "x^2-5;" "6
#,&*$%\"xG\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 52 " that is the fundamenta
l polynomial associated with " }{XPPEDIT 18 0 "sqrt(5);" "6#-%%sqrtG6#
\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 57 " Without goin
g any further, let me simply remark that " }{XPPEDIT 18 0 "x^4-10*x^2+
1" "6#,(*$%\"xG\"\"%\"\"\"*&\"#5F'*$F%\"\"#F'!\"\"F'F'" }{TEXT -1 81 "
doesn't factor as the product of two polynomials in a similar fashion
(and thus " }{XPPEDIT 18 0 "sqrt(2)+sqrt(3)" "6#,&-%%sqrtG6#\"\"#\"\"
\"-F%6#\"\"$F(" }{TEXT -1 17 " is of degree 4):" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "factor(x^4 - 25);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "factor(x^4 - 10*x^2 + 1);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 69 "Elementary properties of algebraic numbers (which ar
e easily proved)." }}{PARA 0 "" 0 "" {TEXT -1 6 "1. If " }{XPPEDIT 18
0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta;" "6#
%%betaG" }{TEXT -1 39 " are algebraic numbers then so too are " }
{XPPEDIT 18 0 "alpha+beta,alpha-beta,alpha*beta;" "6%,&%&alphaG\"\"\"%
%betaGF%,&F$F%F&!\"\"*&F$F%F&F%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a
lpha/beta;" "6#*&%&alphaG\"\"\"%%betaG!\"\"" }{TEXT -1 2 " (" }
{XPPEDIT 18 0 "beta <> 0;" "6#0%%betaG\"\"!" }{TEXT -1 1 ")" }}{PARA
0 "" 0 "" {TEXT -1 6 "2. If " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 43 " is algebraic then so also is every one of " }{XPPEDIT
18 0 "alpha^(1/2),alpha^(1/3),alpha^(1/4),`...`;" "6&)%&alphaG*&\"\"\"
F&\"\"#!\"\")F$*&F&F&\"\"$F()F$*&F&F&\"\"%F(%$...G" }{TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 10 "__________" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT 589 25 "Concerning Hardy & Wright" }
{TEXT -1 3 "'s " }{TEXT 588 71 "It is not immediately obvious that the
re are any transcendental numbers" }{TEXT -1 3 "..." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Why was it natural at o
ne time to wonder about the possibility that " }{TEXT 576 5 "every" }
{TEXT -1 158 " real (or complex) number is algebraic? (Of course one h
as to put out of one's mind the world view after Cantor, with his clar
ification that there are only a " }{TEXT 607 9 "countable" }{TEXT -1
37 " number of algebraic numbers, but an " }{TEXT 605 11 "uncountable
" }{TEXT -1 147 " number of non-algebraic (=transcendental) ones. The \+
interested reader might wish to consult at my web site my 2nd year BA \+
summary course notes on " }{TEXT 606 41 "The Real Numbers and Cantoria
n Set Theory" }{TEXT -1 3 ".) " }}{PARA 0 "" 0 "" {TEXT -1 986 " So
, imagine starting with a completely empty complex plane, and imagine \+
it being filled up, bit by bit, with every kind of number one can thin
k of... every time one thinks of a new number or numbers (especially a
n infinite number of numbers) a light comes on at each of those number
s. The plane begins to be lit up with those tiny point specks of light
... slowly at first... The integers, even all of them, hardly make any
impression, but certainly the rationals fill up a lot of the real lin
e (itself, of course, swamped by the vastness of the entire complex pl
ane), though not, of course, the entire real line: there are all those
real irrational numbers... all those ones that are algebraic of the s
econd degree, but they don't fill the entire real line, there are all \+
those of the third, fourth, fifth, sixth, ... degrees (Do they - perha
ps - fill the entire real line?). And then there are all those complex
numbers with non-zero imaginary parts... all those ones of the form (
" }{XPPEDIT 18 0 "r[1]+r[2]*i;" "6#,&&%\"rG6#\"\"\"F'*&&F%6#\"\"#F'%\"
iGF'F'" }{TEXT -1 9 "), where " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"
\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r[2];" "6#&%\"rG6#\"\"#" }
{TEXT -1 468 " vary in all possible ways over the rational numbers... \+
Every one of those is a solution of a quadratic equation with integer \+
coefficients (and so is of degree 2). They fill up a lot of the comple
x plane, but not, of course, all of it. But then we imagine switching \+
on lights at all the complex solutions of third, fourth, fifth, sixth,
... degree algebraic numbers. Have we now filled up the entire comple
x plane; or, at the very least, earlier, the entire real line?" }}
{PARA 0 "" 0 "" {TEXT -1 13 " Suppose, " }{TEXT 579 19 "just for th
e moment" }{TEXT -1 111 ", that we hadn't filled the entire complex pl
ane, and let us return to the above definition of the actual term " }
{TEXT 577 9 "algebraic" }{TEXT -1 123 ", with its reference to 'intege
r' (or 'rational') coefficients. Let's now make the observation that t
hose coefficients are " }{TEXT 578 6 "merely" }{TEXT -1 95 " (with hin
dsight) algebraic numbers of the first degree, and, in the light of th
at, let us now " }{TEXT 590 9 "speculate" }{TEXT -1 158 " that we migh
t - perhaps - get our hands on a real or complex number that isn't an \+
algebraic number (according to our earlier definition) by doing attemp
ting " }{TEXT 580 4 "like" }{TEXT -1 13 " this: let's " }{TEXT 586 3 "
try" }{TEXT -1 170 " to find a polynomial equation that doesn't have a
n algebraic solution by choosing some (or possibly all) of its coeffic
ients not to be integers (or rational numbers)... " }}{PARA 0 "" 0 ""
{TEXT -1 213 " Note that every such polynomial equation has solutio
ns (as many, in fact, as the degree of the polynomial, thought some ma
y be repeated), every one of which is a complex number: that's because
of the so-called" }{TEXT 581 1 " " }{TEXT 582 30 "Fundamental Theorem
of Algebra" }{TEXT -1 62 " (remarkable itself in its day, and still s
o...), namely that " }{TEXT 584 5 "every" }{TEXT -1 26 " polynomial eq
uation with " }{TEXT 585 12 "real/complex" }{TEXT -1 18 " coefficients
has " }{TEXT 583 29 "exclusively complex solutions" }{TEXT -1 30 " (s
ome of which may be real). " }}{PARA 0 "" 0 "" {TEXT -1 340 " Thus \+
(we wonder) could we find/construct a polynomial, all of whose coeffic
ients are algebraic numbers (but where some/all of them are of degree \+
greater than 1), such that the corresponding equation (all of whose so
lutions we know from the Fundamental Theorem of Algebra must be comple
x numbers) possesses at least one solution that is " }{TEXT 587 3 "not
" }{TEXT -1 141 " an algebraic number? For example, consider the follo
wing equation in which all of the coefficients are algebraic numbers (
not all rational):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "
" {TEXT -1 0 "" }{XPPEDIT 18 0 "sqrt(7)*x^2-9*x/11+3/4-sqrt(5/3) = 0;
" "6#/,**&-%%sqrtG6#\"\"(\"\"\"*$%\"xG\"\"#F*F**(\"\"*F*F,F*\"#6!\"\"F
1*&\"\"$F*\"\"%F1F*-F'6#*&\"\"&F*F3F1F1\"\"!" }{TEXT -1 9 " ... (E)"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "with co
efficients " }{XPPEDIT 18 0 "sqrt(7);" "6#-%%sqrtG6#\"\"(" }{TEXT -1
15 " (2nd degree), " }{XPPEDIT 18 0 "-9/11;" "6#,$*&\"\"*\"\"\"\"#6!\"
\"F(" }{TEXT -1 19 " (1st degree), and " }{XPPEDIT 18 0 "3/4-sqrt(5/3)
;" "6#,&*&\"\"$\"\"\"\"\"%!\"\"F&-%%sqrtG6#*&\"\"&F&F%F(F(" }{TEXT -1
25 " (2nd degree algebraic)." }}{PARA 0 "" 0 "" {TEXT -1 35 "The solu
tions of that equation are:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
50 "solve(sqrt(7)*x^2 - 9*x/11 + 3/4 - sqrt(5/3) = 0);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 65 "A beginner might not immediately notice,
but those two solutions " }{TEXT 595 3 "are" }{TEXT -1 76 " algebraic
numbers according to the standard definition of being algebraic '" }
{TEXT 591 17 "over the integers" }{TEXT -1 2 "'." }}{PARA 0 "" 0 ""
{TEXT -1 386 " Incidentally, it is an entirely elementary matter to
prove - independent of actual calculation - that the two solutions of
the above equation are algebraic numbers (my 2nd year BA students do \+
that sort of thing). In fact, it is an entirely elementary standard re
sult (this is not the place in which to prove it) that no matter what \+
polynomial equation one takes of whatever degree (" }{TEXT 592 1 "m" }
{TEXT -1 117 "), in which all of the coefficients are algebraic number
s of whatever degrees, then not only does that equation have " }{TEXT
593 1 "m" }{TEXT -1 211 " complex solutions (some of which may be real
numbers), possibly with some 'repeated' solutions, but every one of t
hose solutions is an algebraic number (according to the standard defin
ition). In other words, if" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
258 "" 0 "" {XPPEDIT 18 0 "b[0]*x^m+b[1]*x^(m-1)+`...`+b[m] = 0;" "6#/
,**&&%\"bG6#\"\"!\"\"\")%\"xG%\"mGF*F**&&F'6#F*F*)F,,&F-F*F*!\"\"F*F*%
$...GF*&F'6#F-F*F)" }{TEXT -1 9 " ... (B)" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "is a polynomial equation of deg
ree " }{TEXT 594 1 "m" }{TEXT -1 28 " in which every coefficient " }
{XPPEDIT 18 0 "b[0],b[1],`...`,b[m];" "6&&%\"bG6#\"\"!&F$6#\"\"\"%$...
G&F$6#%\"mG" }{TEXT -1 67 " is an algebraic number (according to the s
tandard definition, and " }{XPPEDIT 18 0 "b[0] <> 0;" "6#0&%\"bG6#\"\"
!F'" }{TEXT -1 46 ") then every solution of (B) is a solution of:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "a[0]*x^n+a[1]*x^(n-1)+`...`+a[n] = 0" "6#/,**&&%\"aG6#
\"\"!\"\"\")%\"xG%\"nGF*F**&&F'6#F*F*)F,,&F-F*F*!\"\"F*F*%$...GF*&F'6#
F-F*F)" }{TEXT -1 10 " ... (A)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 27 "in which every coefficient " }{XPPEDIT
18 0 "a[0],a[1],`...`,a[n];" "6&&%\"aG6#\"\"!&F$6#\"\"\"%$...G&F$6#%\"
nG" }{TEXT -1 20 " is an integer, and " }{XPPEDIT 18 0 "a[0] <> 0;" "6
#0&%\"aG6#\"\"!F'" }{TEXT -1 104 ". (The standard way of expressing th
e above succinctly is to say that the field of algebraic numbers is "
}{TEXT 596 20 "algebraically closed" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "
" {TEXT -1 147 " In short, any hope of creating/finding/... a non-a
lgebraic number by merely tinkering with the coefficients (in the mann
er suggested above) is " }{TEXT 598 17 "doomed to failure" }{TEXT -1
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 79 " The question then, in Euler'
s time (or possibly earlier than Euler?), was: " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 599 10 "are there " }{TEXT
282 3 "any" }{TEXT 600 50 " (real or complex) numbers that are not alg
ebraic?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
42 " Euler (or someone earlier?) defined a " }{TEXT 601 21 "transce
ndental number" }{TEXT -1 105 " to be a (real or complex) number that \+
is not algebraic; it \"transcendended the power of the algebraic\". "
}}{PARA 0 "" 0 "" {TEXT -1 14 " Of course " }{TEXT 602 2 "if" }
{TEXT -1 17 " there were any, " }{TEXT 603 4 "then" }{TEXT -1 103 " th
ere would automatically be infinitely many. That's a matter that a nov
ice might like to think about." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "I return to Euler in another subsection.
" }}{PARA 258 "" 0 "" {TEXT -1 29 "_____________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 286 "A final \+
word about computer computation/demonstration. I'd like to return brie
fly to the example (E) above, just in case a novice (using Maple or wh
atever) attempts to vary the coefficients, and wonders about why somet
imes things don't appear to work out nicely... Consider the example:"
}}{PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "sqrt(7)*x^5-9*x/11+3
/4-sqrt(5/3) = 0;" "6#/,**&-%%sqrtG6#\"\"(\"\"\"*$%\"xG\"\"&F*F**(\"\"
*F*F,F*\"#6!\"\"F1*&\"\"$F*\"\"%F1F*-F'6#*&F-F*F3F1F1\"\"!" }{TEXT -1
10 " ... (E')" }}{PARA 0 "" 0 "" {TEXT -1 42 "in which I have merely \+
changed in (E) the " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1
4 " to " }{XPPEDIT 18 0 "x^5;" "6#*$%\"xG\"\"&" }{TEXT -1 79 ", but ot
herwise kept everything else unaltered. Now try 'solving', and you get
:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "solve(sqrt(7)*x^5 - 9*x
/11 + 3/4 - sqrt(5/3) = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 309 "The software appears to fail, an
d with good mathematical reason... Here a beginner needs to know that \+
one has now come up against a barrier: the classical result (Abel and \+
Galois) concerning the non-existence of a solution 'using radicals' fo
r general polynomial equations of degree at least 5... But that is " }
{TEXT 604 23 "\"another day's work\"..." }{TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "You can of course do
things like:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(sqrt(7
)*x^5 - 9*x/11 + 3/4 - sqrt(5/3), x = -2..2);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 56 "plot(sqrt(7)*x^5 - 9*x/11 + 3/4 - sqrt(5/3), x =
-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fsolve(sqrt(7)*x^5 - 9*x/11 + 3/4 -
sqrt(5/3) = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 16 " Enough of this." }}}{SECT 1 {PARA 4 ""
0 "" {TEXT -1 33 "Some well-known algebraic numbers" }}{SECT 1 {PARA
5 "" 0 "" {TEXT 285 24 "Certain quadratics: The " }{TEXT -1 12 "golden
ratio" }{TEXT 832 1 " " }{XPPEDIT 18 0 "(sqrt(5)+1)/2;" "6#*&,&-%%sqr
tG6#\"\"&\"\"\"F)F)F)\"\"#!\"\"" }{TEXT 304 6 ", and " }{XPPEDIT 18 0
"sqrt(2);" "6#-%%sqrtG6#\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 23 "Dan Brown's bestseller " }{TEXT 294 17 "T
he Da Vinci Code" }{TEXT 295 1 " " }{TEXT -1 31 "features one of the m
ost famous" }{TEXT 296 1 " " }{TEXT -1 164 "algebraic numbers (without
, of course, being referred to as such!); unfortunately his presentati
on contains a number of serious errors. I quote from his Chapter 20:"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 344 "\" [La
ngdon] felt himself suddenly reeling back to Harvard, standing in fron
t of his 'Symbolism in Art' class, writing his favourite number on the
chalkboard: 1.618\n Langdon turned to face his sea of eager student
s. 'Who can tell me what this number is?\n A long-legged maths major
at the back of the class raised his hand. 'That's the number" }{TEXT
302 1 " " }{TEXT -1 23 "PHI.' He pronounced it " }{TEXT 300 3 "fee" }
{TEXT -1 62 ".\n 'Nice job, Stettner,' Langdon said. 'Everyone, meet
PHI.'" }}{PARA 0 "" 0 "" {TEXT -1 495 " ... ... \n 'This numbe
r PHI,' Langdon continued, 'one-point-six-one-eight, is a very importa
nt number in art. Who can tell me why?'\n ... ... \n 'Actually,
' Langdon said, ... 'PHI is generally considered the most beautiful nu
mber in the universe.' \n ... ...\n As Langdon loaded his slide \+
projector, he explained that the number PHI was derived from the Fibon
acci sequence - a progression famous not only because the sum of adjac
ent terms equalled the next term, but because the " }{TEXT 301 9 "quot
ients" }{TEXT -1 128 " [Brown's italics] of adjacent terms possessed t
he astonishing property of approaching the number 1.618 - PHI! \" [end
of quote] " }}{PARA 0 "" 0 "" {TEXT -1 5 " '" }{XPPEDIT 18 0 "phi;
" "6#%$phiG" }{TEXT -1 23 "' (PHI), the so-called " }{TEXT 299 12 "gol
den ratio" }{TEXT -1 16 ", is the number " }{XPPEDIT 18 0 "(sqrt(5)+1)
/2;" "6#*&,&-%%sqrtG6#\"\"&\"\"\"F)F)F)\"\"#!\"\"" }{TEXT -1 60 ", and
is one of the two solutions of the quadratic equation " }{XPPEDIT 18
0 "x^2-x-1 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"F&!\"\"F(F)\"\"!" }{TEXT -1
127 ". This number is so well known that it would be foolish for me to
write about it at length; however I will make a few comments." }}
{PARA 0 "" 0 "" {TEXT -1 11 " First, " }{XPPEDIT 18 0 "phi;" "6#%$p
hiG" }{TEXT -1 22 "is NOT \"1.618\". It is " }{TEXT 303 19 "entirely e
lementary" }{TEXT -1 15 " to argue that " }{XPPEDIT 18 0 "(sqrt(5)+1)/
2;" "6#*&,&-%%sqrtG6#\"\"&\"\"\"F)F)F)\"\"#!\"\"" }{TEXT -1 220 " cann
ot have a terminating decimal expansion, and not only that, but being \+
irrational it cannot have a periodic decimal expansion. (Of course I c
an see that Brown could have put some readers off their coffee had he \+
used " }{XPPEDIT 18 0 "(sqrt(5)+1)/2" "6#*&,&-%%sqrtG6#\"\"&\"\"\"F)F)
F)\"\"#!\"\"" }{TEXT -1 189 " ... , but surely he could have been advi
sed to use '1.618... ' What view could readers form of mathematicians \+
who would 'consider' 1.618 to be 'the most beautiful number in the uni
verse'?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 24 "solve(x^2 - x - 1 = 0);\n" }{TEXT -1 0 "" }{MPLTEXT
1 0 26 "alpha := (sqrt(5) + 1)/2;\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 13 "
evalf(alpha);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "It is well-known that " }{XPPEDIT
18 0 "phi;" "6#%$phiG" }{TEXT -1 40 " is the limit of the infinite seq
uence \{" }{XPPEDIT 18 0 "F[n+1]/F[n];" "6#*&&%\"FG6#,&%\"nG\"\"\"F)F)
F)&F%6#F(!\"\"" }{TEXT -1 9 "\}, where " }{XPPEDIT 18 0 "F[n];" "6#&%
\"FG6#%\"nG" }{TEXT -1 8 " is the " }{TEXT 831 1 "n" }{TEXT -1 75 "-th
Fibonacci number (they, too, make an appearance in the Da Vinci Code)
. " }}{PARA 0 "" 0 "" {TEXT -1 80 "The Fibonacci numbers are the terms
of the sequence 1, 1, 2, 3, 5, 8, 13, ... , " }{XPPEDIT 18 0 "F[n];"
"6#&%\"FG6#%\"nG" }{TEXT -1 14 ", ... , where " }{XPPEDIT 18 0 "F[1] =
1,F[2] = 1;" "6$/&%\"FG6#\"\"\"F'/&F%6#\"\"#F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "F[n] = F[n-1]+F[n-2];" "6#/&%\"FG6#%\"nG,&&F%6#,&F'\"\"
\"F,!\"\"F,&F%6#,&F'F,\"\"#F-F," }{TEXT -1 5 " for " }{XPPEDIT 18 0 "3
<= n;" "6#1\"\"$%\"nG" }{TEXT -1 32 ". Here are the first 25 of them:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "F[1] := 1: F[2] := 1: for k from 3 \+
to 25 do\nF[k] := F[k-1]+F[k-2] od:\nseq(F[k], k = 1..25);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 71 "It is an elementary exercise for a beginner to prove that the s
equence " }{XPPEDIT 18 0 "F[2]/F[1],F[4]/F[3],F[6]/F[5],F[8]/F[7],`...
`,F[2*n]/F[2*n-1],`...`;" "6)*&&%\"FG6#\"\"#\"\"\"&F%6#F(!\"\"*&&F%6#
\"\"%F(&F%6#\"\"$F+*&&F%6#\"\"'F(&F%6#\"\"&F+*&&F%6#\"\")F(&F%6#\"\"(F
+%$...G*&&F%6#*&F'F(%\"nGF(F(&F%6#,&*&F'F(FFF(F(F(F+F+FA" }{TEXT -1
23 " is strictly monotonic " }{TEXT 847 2 "in" }{TEXT -1 58 "creasing \+
(early terms followed by successive differences):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "seq(F[2*n]/F[2*n-1], n = 1..12);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(F[2*n+2]/F[2*n+1] - F[2*n]/F[2*
n-1], n = 1..11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 " while the sequence " }{XPPEDIT
18 0 "F[3]/F[2],F[5]/F[4],F[7]/F[6],F[9]/F[8],`...`,F[2*n+1]/F[2*n],`.
..`;" "6)*&&%\"FG6#\"\"$\"\"\"&F%6#\"\"#!\"\"*&&F%6#\"\"&F(&F%6#\"\"%F
,*&&F%6#\"\"(F(&F%6#\"\"'F,*&&F%6#\"\"*F(&F%6#\"\")F,%$...G*&&F%6#,&*&
F+F(%\"nGF(F(F(F(F(&F%6#*&F+F(FHF(F,FB" }{TEXT -1 23 " is strictly mon
otonic " }{TEXT 848 2 "de" }{TEXT -1 10 "creasing: " }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 32 "seq(F[2*n+1]/F[2*n], n = 1..12);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "seq(F[2*n+3]/F[2*n+2] - F[2*
n+1]/F[2*n], n = 1..11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Also, the " }{TEXT 849 2 "n-
" }{TEXT -1 47 "th term of the first sequence is less that the " }
{TEXT 850 2 "n-" }{TEXT -1 82 "th term of the second sequence, and the
difference between those tends to zero as " }{XPPEDIT 18 0 "proc (n) \+
options operator, arrow; infinity end proc;" "6#R6#%\"nG7\"6$%)operato
rG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "seq(F[2*n+1]/F[2*n] - F[2*n]/F[2*n-1], n = 1..
11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "Limit(F[n+1]/F[n],n = infin
ity);" "6#-%&LimitG6$*&&%\"FG6#,&%\"nG\"\"\"F,F,F,&F(6#F+!\"\"/F+%)inf
inityG" }{TEXT -1 35 " exists, and is easily shown to be " }{XPPEDIT
18 0 "(sqrt(5)+1)/2;" "6#*&,&-%%sqrtG6#\"\"&\"\"\"F)F)F)\"\"#!\"\"" }
{TEXT -1 10 ", namely '" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1
12 "'. In fact, " }{TEXT 852 5 "every" }{TEXT -1 24 " term of the sequ
ence \{ " }{XPPEDIT 18 0 "F[n+1]/F[n];" "6#*&&%\"FG6#,&%\"nG\"\"\"F)F)
F)&F%6#F(!\"\"" }{TEXT -1 13 " \}, from the " }{TEXT 853 5 "12-th" }
{TEXT -1 13 " onwards, is " }{TEXT 851 7 "greater" }{TEXT -1 29 " than
Brown's 1.618 'limit': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35
"seq(evalf(F[k+1]/F[k]), k = 1..24);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 18 "In short, just as "
}{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 18 " is not 3.14 (nor " }
{XPPEDIT 18 0 "22/7;" "6#*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 3 "), " }
{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 14 " is not 1.618." }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 287 22 "Complex roots of \+
unity" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111
"School pupils familiar with complex numbers and DeMoivre's theorem kn
ow examples of algebraic numbers. For let " }{XPPEDIT 18 0 "alpha = co
s(2*Pi/n)+i*sin(2*Pi/n);" "6#/%&alphaG,&-%$cosG6#*(\"\"#\"\"\"%#PiGF+%
\"nG!\"\"F+*&%\"iGF+-%$sinG6#*(F*F+F,F+F-F.F+F+" }{TEXT -1 8 " (where \+
" }{TEXT 536 1 "n" }{TEXT -1 29 " is any natural number), then" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "alpha
^n = (cos(2*Pi/n)+i*sin(2*Pi/n))^n;" "6#/)%&alphaG%\"nG),&-%$cosG6#*(
\"\"#\"\"\"%#PiGF.F&!\"\"F.*&%\"iGF.-%$sinG6#*(F-F.F/F.F&F0F.F.F&" }
{TEXT -1 2 "= " }{XPPEDIT 18 0 "cos(n*2*Pi/n)+i*sin(n*2*Pi/n) = cos(2*
Pi)+i*sin(2*Pi);" "6#/,&-%$cosG6#**%\"nG\"\"\"\"\"#F*%#PiGF*F)!\"\"F**
&%\"iGF*-%$sinG6#**F)F*F+F*F,F*F)F-F*F*,&-F&6#*&F+F*F,F*F**&F/F*-F16#*
&F+F*F,F*F*F*" }{TEXT -1 4 " = 1" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "and thus " }{XPPEDIT 18 0 "x = alpha;" "6#
/%\"xG%&alphaG" }{TEXT -1 42 " is a solution of the polynomial equatio
n " }{XPPEDIT 18 0 "x^n-1 = 0;" "6#/,&)%\"xG%\"nG\"\"\"F(!\"\"\"\"!" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 13 "In fact, the " }{TEXT 542 1 "n" }{TEXT -1 27 " solutions \+
of the equation " }{XPPEDIT 18 0 "x^n-1 = 0" "6#/,&)%\"xG%\"nG\"\"\"F(
!\"\"\"\"!" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "alpha,alpha^2,`...`,al
pha^(n-1),alpha^n;" "6'%&alphaG*$F#\"\"#%$...G)F#,&%\"nG\"\"\"F*!\"\")
F#F)" }{TEXT -1 6 " (=1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "n = 6;" "6#/%
\"nG\"\"'" }{TEXT -1 15 ", the equation " }{XPPEDIT 18 0 "x^n-1 = 0;"
"6#/,&)%\"xG%\"nG\"\"\"F(!\"\"\"\"!" }{TEXT -1 36 " has 6 (algebraic n
umber) solutions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 ""
{TEXT -1 32 "one of which will be of degree 1" }}{PARA 15 "" 0 ""
{TEXT -1 41 "another of which will also be of degree 1" }}{PARA 15 ""
0 "" {TEXT -1 32 "two of which will be of degree 2" }}{PARA 15 "" 0 "
" {TEXT -1 49 "and another two of which will also be of degree 2" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n := 6:\nfactor(x^n - 1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 27 "whereas, for example, when " }{XPPEDIT 18 0 "n = 7;" "6#/%\"nG
\"\"(" }{TEXT -1 70 ", something quite different will happen (and I wi
ll not elaborate...):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n :
= 7:\nfactor(x^n - 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 289 21 "Certain value
s of the" }{TEXT -1 1 " " }{TEXT 608 2 "j(" }{TEXT -1 1 "z" }{TEXT
613 27 ")-elliptic modular function" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }
}{PARA 5 "" 0 "" {TEXT 917 14 "The remarkable" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "j(z);" "6#-%\"jG6#%\"zG" }{TEXT -1 27 "-elliptic modula
r function " }{TEXT 609 24 "of the complex variable " }{TEXT 855 2 "z \+
" }{TEXT -1 1 "(" }{TEXT 610 4 "with" }{TEXT -1 2 " " }{TEXT 611 3 "I
m(" }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT 612 52 ") > 0 ) is defined in
many ways, but one of them is:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "j(
z) = (1+240*Sum(sigma[3](n)*e^(2*Pi*i*n*z),n = 1 .. infinity))^3/(e^(2
*Pi*i*z)*Product(1-e^(2*Pi*i*n*z),n = 1 .. infinity)^24);" "6#/-%\"jG6
#%\"zG*&,&\"\"\"F**&\"$S#F*-%$SumG6$*&-&%&sigmaG6#\"\"$6#%\"nGF*)%\"eG
*,\"\"#F*%#PiGF*%\"iGF*F7F*F'F*F*/F7;F*%)infinityGF*F*F5*&)F9**F;F*F " 0
"" {MPLTEXT 1 0 67 "p := 163;\nx := round(exp(Pi*sqrt(p)/3));\ny := sq
rt((x^3 + 1728)/p);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Inciden
tally a 1913 theorem of Rabinovitch (see, e.g., Goldfeld's 1985 AMS "
}{TEXT 877 8 "Bulletin" }{TEXT -1 40 " article) intimately relates the
above '" }{TEXT 878 1 "p" }{TEXT -1 118 "' primes to remarkable prime
-producing polynomials. Almost everyone has encountered the discovered
-by-Euler quadratic " }{XPPEDIT 18 0 "x^2-x+41;" "6#,(*$%\"xG\"\"#\"\"
\"F%!\"\"\"#TF'" }{TEXT -1 45 ", which is prime for every integral val
ue of " }{TEXT 879 1 "x" }{TEXT -1 90 " between 1 and 40 inclusive. Th
e '41' is tied - by Rabinovitch's theorem to the '163' via " }
{XPPEDIT 18 0 "41 = (163+1)/4;" "6#/\"#T*&,&\"$j\"\"\"\"F(F(F(\"\"%!\"
\"" }{TEXT -1 124 ", and indeed the other primes above (7, 11, 19, 43 \+
and 67) produce similar prime-producing polynomials in identical fashi
on." }}{PARA 0 "" 0 "" {TEXT -1 57 "The final ten prime values in the \+
case of the polynomial " }{XPPEDIT 18 0 "x^2-x+41;" "6#,(*$%\"xG\"\"#
\"\"\"F%!\"\"\"#TF'" }{TEXT -1 34 " ('live' I can show all of them): \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for x from 31 to 40 do\n
print([x^2 - x + 41, isprime(x^2 - x + 41)])\nod;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and \+
all of them for the " }{TEXT 880 4 "p = " }{TEXT -1 2 "67" }{TEXT
1253 1 " " }{TEXT 881 0 "" }{TEXT -1 15 "example, where " }{XPPEDIT
18 0 "q = (p+1)/4;" "6#/%\"qG*&,&%\"pG\"\"\"F(F(F(\"\"%!\"\"" }{TEXT
-1 14 " = 17, giving " }{XPPEDIT 18 0 "x^2-x+17;" "6#,(*$%\"xG\"\"#\"
\"\"F%!\"\"\"# " 0 ""
{MPLTEXT 1 0 63 "for x to 16 do\nprint([x^2 - x + 17, isprime(x^2 - x \+
+ 17)])\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA
0 "" 0 "" {TEXT -1 28 "Here are two j-values: " }}{PARA 258 "" 0
"" {TEXT -1 11 "\n " }{XPPEDIT 18 0 "j((1+sqrt(-67))/2) = (-5
280)^3;" "6#/-%\"jG6#*&,&\"\"\"F)-%%sqrtG6#,$\"#n!\"\"F)F)\"\"#F/*$,$
\"%!G&F/\"\"$" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "j((1+sqrt(-163))/
2) = (-640320)^3;" "6#/-%\"jG6#*&,&\"\"\"F)-%%sqrtG6#,$\"$j\"!\"\"F)F)
\"\"#F/*$,$\"'?.kF/\"\"$" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 118 "Finally I remark that it had been a
long-standing question - going back as far as Gauss (in connection wi
th his wider " }{TEXT 882 20 "class-number problem" }{TEXT -1 81 ") - \+
as to whether or not 163 was the largest prime for which the class num
ber of " }{XPPEDIT 18 0 "Q(sqrt(-p));" "6#-%\"QG6#-%%sqrtG6#,$%\"pG!\"
\"" }{TEXT -1 85 " is 1; equivalently, because of Rabinovitch, whether
or not 41 was the largest prime " }{TEXT 883 1 "q" }{TEXT -1 3 " (="
}{XPPEDIT 18 0 "(p+1)/4;" "6#*&,&%\"pG\"\"\"F&F&F&\"\"%!\"\"" }{TEXT
-1 12 ") such that " }{XPPEDIT 18 0 "x^2-x+q;" "6#,(*$%\"xG\"\"#\"\"\"
F%!\"\"%\"qGF'" }{TEXT -1 18 " is prime for all " }{TEXT 884 1 "x" }
{TEXT -1 15 " between 1 and " }{XPPEDIT 18 0 "q-1;" "6#,&%\"qG\"\"\"F%
!\"\"" }{TEXT -1 146 ". That question was answered in the affirmative \+
by Heegner, Baker, and Stark in the early 50s and mid 60s. There the h
istory is truly complicated." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}
{PARA 0 "" 0 "" {TEXT -1 113 "Finally, today, the 6th of May 2004, is \+
the 50th anniversary of the breaking by Roger Bannister in Oxford of t
he " }{TEXT 1313 24 "four-minute mile barrier" }{TEXT -1 110 ". I can'
t resist pointing out that all the numerical omens were just right for
him on that history-making day:" }}{PARA 15 "" 0 "" {TEXT -1 53 "'6' \+
- as in May 6th - is as everyone knows the first " }{TEXT 1314 7 "perf
ect" }{TEXT -1 7 " number" }}{PARA 15 "" 0 "" {TEXT -1 239 "The number
Bannister wore on his vest on that memorable day was '41', which feat
ures above in the - what one might call - prime record breaking polyno
mial, and the '41' ties in with the '163' in the final one of the abov
e modular equations" }}{PARA 15 "" 0 "" {TEXT -1 117 "The '5280' which
features in the previous one - the one '41' surpassed and set a recor
d - is of course the number of " }{TEXT 1315 4 "feet" }{TEXT -1 83 " i
n a mile (a fact perhaps lost on those of you who grew up with the met
ric system)" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 286 38 "C
onway's Look-and-Say algebraic number" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 120 "Perhaps the most remarkable example \+
of an algebraic number (from a gee-whiz point of view) is provided by \+
John Conway's " }{TEXT 290 4 "Look" }{TEXT -1 5 "-and-" }{TEXT 291 3 "
Say" }{TEXT -1 29 " sequence (see the wonderful " }{TEXT 292 15 "Book \+
of Numbers" }{TEXT -1 18 " by Conway & Guy):" }}{PARA 258 "" 0 ""
{TEXT -1 75 "1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31
131211121221, ... " }}{PARA 0 "" 0 "" {TEXT -1 290 " I hope you see
why it's called the Look-and-Say sequence... (If you don't already kn
ow of N. J. A. Sloane's extraordinary On-Line Encyclopedia of Integer \+
Sequences, then you simply must consult it; be prepared, however, to s
uffer the disappointment of thinking you've discovered a new (" }
{TEXT 293 11 "interesting" }{TEXT -1 56 ") sequence: you will almost c
ertainly find it there...)." }}{PARA 0 "" 0 "" {TEXT -1 29 " How ma
ny digits does the " }{TEXT 918 1 "n" }{TEXT -1 81 "-th term of that \+
sequence have? Conway proved that it's roughly proportional to " }
{XPPEDIT 18 0 "alpha^n;" "6#)%&alphaG%\"nG" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "alpha = 1.303577269034296391257099112152551890730702504
6594*`...`;" "6#/%&alphaG*&$\"S%fY]-2t!*=b_@6*4d7R'HM!psd.8!#\\\"\"\"%
$...GF)" }{TEXT -1 85 ", is an algebraic number of degree 71, being a \+
solution of the following irreducible " }{XPPEDIT 18 0 "71^st;" "6#)\"
#r%#stG" }{TEXT -1 56 " degree polynomial equation (with integer coeff
icients):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
1 " " }{XPPEDIT 18 0 "x^71-x^69-2*x^68-x^67+2*x^66+2*x^65+x^64-x^63-x^
62-x^61-x^60-x^59+2*x^58+5*x^57+3*x^56-2*x^55-10*x^54-3*x^53-2*x^52+6*
x^51+6*x^50+x^49+9*x^48-3*x^47-7*x^46-8*x^45-8*x^44+10*x^43+6*x^42+8*x
^41-5*x^40-12*x^39+7*x^38-7*x^37+7*x^36+x^35-3*x^34+10*x^33+x^32-6*x^3
1-2*x^30-10*x^29-3*x^28+2*x^27+9*x^26-3*x^25+14*x^24-8*x^23-7*x^21+9*x
^20+3*x^19-4*x^18-10*x^17-7*x^16+12*x^15+7*x^14+2*x^13-12*x^12-4*x^11-
2*x^10+5*x^9+x^7-7*x^6+7*x^5-4*x^4+12*x^3-6*x^2+3*x-6 = 0;" "6#/,fs*$%
\"xG\"#r\"\"\"*$F&\"#p!\"\"*&\"\"#F(*$F&\"#oF(F+*$F&\"#nF+*&F-F(*$F&\"
#mF(F(*&F-F(*$F&\"#lF(F(*$F&\"#kF(*$F&\"#jF+*$F&\"#iF+*$F&\"#hF+*$F&\"
#gF+*$F&\"#fF+*&F-F(*$F&\"#eF(F(*&\"\"&F(*$F&\"#dF(F(*&\"\"$F(*$F&\"#c
F(F(*&F-F(*$F&\"#bF(F+*&\"#5F(*$F&\"#aF(F+*&FLF(*$F&\"#`F(F+*&F-F(*$F&
\"#_F(F+*&\"\"'F(*$F&\"#^F(F(*&FgnF(*$F&\"#]F(F(*$F&\"#\\F(*&\"\"*F(*$
F&\"#[F(F(*&FLF(*$F&\"#ZF(F+*&\"\"(F(*$F&\"#YF(F+*&\"\")F(*$F&\"#XF(F+
*&F[pF(*$F&\"#WF(F+*&FSF(*$F&\"#VF(F(*&FgnF(*$F&\"#UF(F(*&F[pF(*$F&\"#
TF(F(*&FHF(*$F&\"#SF(F+*&\"#7F(*$F&\"#RF(F+*&FgoF(*$F&\"#QF(F(*&FgoF(*
$F&\"#PF(F+*&FgoF(*$F&\"#OF(F(*$F&\"#NF(*&FLF(*$F&\"#MF(F+*&FSF(*$F&\"
#LF(F(*$F&\"#KF(*&FgnF(*$F&\"#JF(F+*&F-F(*$F&\"#IF(F+*&FSF(*$F&\"#HF(F
+*&FLF(*$F&\"#GF(F+*&F-F(*$F&\"#FF(F(*&F`oF(*$F&\"#EF(F(*&FLF(*$F&\"#D
F(F+*&\"#9F(*$F&\"#CF(F(*&F[pF(*$F&\"#BF(F+*&FgoF(*$F&\"#@F(F+*&F`oF(*
$F&\"#?F(F(*&FLF(*$F&\"#>F(F(*&\"\"%F(*$F&\"#=F(F+*&FSF(*$F&\"# " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 607 "f := x -> x^71 - x^69 - 2*x^68 - x^67 + 2*x^66 + 2*x
^65 + x^64 \n- x^63 - x^62 - x^61 - x^60 - x^59 + 2*x^58 + 5*x^57 + 3*
x^56 \n- 2*x^55 - 10*x^54 - 3*x^53 - 2*x^52 + 6*x^51 + 6*x^50 + x^49 \+
\n+ 9*x^48 - 3*x^47 - 7*x^46 - 8*x^45 - 8*x^44 + 10*x^43 + 6*x^42 \n+ \+
8*x^41 - 5*x^40 - 12*x^39 + 7*x^38 - 7*x^37 + 7*x^36 + x^35 \n- 3*x^34
+ 10*x^33 + x^32 - 6*x^31 - 2*x^30 - 10*x^29 - 3*x^28 \n+ 2*x^27 + 9*
x^26 - 3*x^25 + 14*x^24 - 8*x^23 - 7*x^21 + 9*x^20 \n+ 3*x^19 - 4*x^18
- 10*x^17 - 7*x^16 + 12*x^15 + 7*x^14 + 2*x^13 \n- 12*x^12 - 4*x^11 -
2*x^10 + 5*x^9 + x^7 - 7*x^6 + 7*x^5 - 4*x^4 \n+ 12*x^3 - 6*x^2 + 3*x
- 6:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Digits := 50; # to
give the value in Conway & Guy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 27 "fsolve(f(x) = 0, x = 1..2);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "Digits := 10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 33 "plot(f(x), x = (1.303)..(1.304));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Euler's \+
intuition (which later proved correct)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 5 "From " }{TEXT 843 12 "A.O. Gelfond" }
{TEXT -1 11 "'s classic " }{TEXT 841 47 "Transcendental and Algebraic \+
Numbers of Numbers" }{TEXT -1 52 " (English translation of Russian ori
ginal) I quote: " }{TEXT 842 406 "The Euler-Hilbert problem. The probl
em of the transcendence or the rationality of the logarithms, with rat
ional base, of rational numbers, stated by Euler in 1748, was formulat
ed by Hilbert in a significantly more general form and introduced by h
im as number seven of a set of 23 problems, to the solution of which t
here appeared to be no suitable approach even at the very end of the n
ineteenth century..." }{TEXT 844 1 " " }{TEXT -1 47 "(I take up this t
opic in greater detail later.)" }}{PARA 0 "" 0 "" {TEXT -1 185 " Eu
ler introduced the concept of a transcendental number (When exactly? W
as it done before Euler? Was it Leibnitz? Is there some confusion over
the difference between transcendental " }{TEXT 845 8 "function" }
{TEXT -1 20 " and transcendental " }{TEXT 846 6 "number" }{TEXT -1 17
"?), but he could " }{TEXT 307 3 "not" }{TEXT -1 28 " prove the transc
endence of " }{TEXT 305 13 "even a single" }{TEXT -1 59 " example (one
imagines that he must have wondered if (e.g) " }{XPPEDIT 18 0 "Pi;" "
6#%#PiG" }{TEXT -1 4 " or " }{TEXT 306 1 "e" }{TEXT -1 117 " were tran
scendental). Euler did however make a wonderful guess (Fel'dman and Sh
idlovskii - in their monumental 1967 " }{TEXT 614 6 "Survey" }{TEXT
-1 8 " wrote: " }{TEXT 615 286 "\"...we may mention the conjecture mad
e by Euler in 1748 (they give as reference Euler's Introductio in anal
ysin infinitorum, Lausanne 1748, Opera omnia, VIII and IX) on the tran
scendence of the logarithms to a rational base of rational numbers tha
t are not rational powers of the base.\"" }{TEXT -1 144 ") at some num
bers that he felt could be transcendental (it is an elementary exercis
e that his candidates are all irrational), as I now explain. " }}
{PARA 0 "" 0 "" {TEXT -1 54 " Standard, routine high school problem
s are to ask:" }}{PARA 15 "" 0 "" {TEXT -1 19 "Solve the equation " }
{XPPEDIT 18 0 "4^x = 8;" "6#/)\"\"%%\"xG\"\")" }{TEXT -1 47 " (later v
ary the pair (4, 8) to e.g: (4, 16), (" }{XPPEDIT 18 0 "27/8;" "6#*&\"
#F\"\"\"\"\")!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "9/4;" "6#*&\"\"*
\"\"\"\"\"%!\"\"" }{TEXT -1 4 "), (" }{XPPEDIT 18 0 "1/100;" "6#*&\"\"
\"F$\"$+\"!\"\"" }{TEXT -1 12 ", 10), ... )" }}{PARA 15 "" 0 "" {TEXT
-1 19 "Solve the equation " }{XPPEDIT 18 0 "4^x = 6;" "6#/)\"\"%%\"xG
\"\"'" }{TEXT -1 55 " (later vary the pair (4, 6) to e.g: (5, 2), (10,
8), (" }{XPPEDIT 18 0 "11/3;" "6#*&\"#6\"\"\"\"\"$!\"\"" }{TEXT -1 2
", " }{XPPEDIT 18 0 "12/7;" "6#*&\"#7\"\"\"\"\"(!\"\"" }{TEXT -1 8 "),
... )" }}{PARA 259 "" 0 "" {TEXT -1 50 "\nOf course both equations ha
ve solutions, meaning:" }}{PARA 15 "" 0 "" {TEXT -1 6 "there " }{TEXT
308 2 "is" }{TEXT -1 5 " an '" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"
\"" }{TEXT -1 12 "' such that " }{XPPEDIT 18 0 "4^x[1] = 8;" "6#/)\"\"
%&%\"xG6#\"\"\"\"\")" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 6 "t
here " }{TEXT 309 2 "is" }{TEXT -1 5 " an '" }{XPPEDIT 18 0 "x[2];" "6
#&%\"xG6#\"\"#" }{TEXT -1 12 "' such that " }{XPPEDIT 18 0 "4^x[2] = 6
;" "6#/)\"\"%&%\"xG6#\"\"#\"\"'" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 ""
{TEXT -1 9 "But what " }{TEXT 310 3 "are" }{TEXT -1 15 " the values of
" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 74 "? School pupils \+
- spotting the obviously related '4' and '8' - should get " }{XPPEDIT
18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "
3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 35 ", but might get stuck \+
with finding " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 34 "
. Some might say something like: " }{XPPEDIT 18 0 "x[2]*log[10](4) = \+
log[10](6);" "6#/*&&%\"xG6#\"\"#\"\"\"-&%$logG6#\"#56#\"\"%F)-&F,6#F.6
#\"\"'" }{TEXT -1 36 ", then - using a calculator - write:" }}{PARA
258 "" 0 "" {XPPEDIT 18 0 "x[2] = log[10](6)/log[10](4);" "6#/&%\"xG6#
\"\"#*&-&%$logG6#\"#56#\"\"'\"\"\"-&F+6#F-6#\"\"%!\"\"" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 ".7781512504/.6020599913 = 1.292481250;" "6#/*&$\"+
/D^\"y(!#5\"\"\"$\"+8**f?gF'!\"\"$\"+]7[#H\"!\"*" }{TEXT -1 27 " (to 1
0 places of accuracy)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 40 "It is entirely elementary to prove that " }{XPPEDIT
18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 62 " is, in fact, irrational.
One might be astounded to know that " }{XPPEDIT 18 0 "x[2];" "6#&%\"x
G6#\"\"#" }{TEXT -1 4 " is " }{TEXT 311 9 "actually " }{TEXT -1 15 "tr
anscendental." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT 312 37 "This is what I believe Euler surmised" }{TEXT -1 6 ": le
t " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "r[2];" "6#&%\"rG6#\"\"#" }{TEXT -1 40 " be positive rat
ional numbers such that " }{XPPEDIT 18 0 "r[1]^p = r[2];" "6#/)&%\"rG6
#\"\"\"%\"pG&F&6#\"\"#" }{TEXT -1 6 ", then" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 313 1 "p" }{TEXT 315
13 " is rational " }{TEXT -1 2 "OR" }{TEXT 318 1 " " }{TEXT 314 1 "p"
}{TEXT 316 18 " is transcendental" }{TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "(Alternatively: the ratio
of the logarithms of two rational numbers " }{XPPEDIT 18 0 "log(r[2])
/log(r[1]);" "6#*&-%$logG6#&%\"rG6#\"\"#\"\"\"-F%6#&F(6#F+!\"\"" }
{TEXT -1 39 " is either rational or transcendental.)" }}{PARA 0 "" 0 "
" {TEXT -1 21 "One should sense the " }{TEXT 320 17 "remarkable nature
" }{TEXT -1 91 " of this assertion/guess/conjecture... it is saying/su
ggesting that such a ratio is either " }{TEXT 317 17 "incredibly simpl
e" }{TEXT -1 4 " or " }{TEXT 319 18 "incredibly complex" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
572 "with(plots): # for 'display'\nwith(plottools): # for 'line'\np :=
plot(4^x, x = 0.9..1.6, colour=navy, thickness=2,\n title
=\"THE GRAPH OF 4 TO THE POWER x\"):\nv1 := line([1.5, 0], [1.5, 8], c
olor=red, thickness=2):\nalpha := log[10](6.0)/log[10](4.0):\nv2 := li
ne([alpha, 0], [alpha, 6], colour=brown, thickness=2):\ntp1 := textplo
t([1.48, 8,`x[1]=3/2`],align=LEFT):\ntp2 := textplot([1.51, 8,`y=8`],a
lign=RIGHT):\ntp3 := textplot([1.3, 6,`y=6`],align=RIGHT):\ntp4 := tex
tplot([1.28, 6.2,`Is x[2]=transcendental?`],align=LEFT):\ndisplay([p, \+
v1, v2, tp1, tp2, tp3, tp4]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
84 "Hilbert's seventh problem (later section) is a more general form o
f Euler's surmise." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA
3 "" 0 "" {TEXT -1 37 "Liouville, Cantor, Hermite, Lindemann" }}{SECT
1 {PARA 4 "" 0 "" {TEXT -1 21 "Liouville (1809-1882)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 544 0 "" }
{TEXT 545 42 "What is now proved was once only imagin'd " }{TEXT -1
15 "(William Blake)" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 128 "All authors credit Liouville with being the discovere
r (creator?) of the first example of a transcendental number, but ther
e is " }{TEXT 409 18 "definite confusion" }{TEXT -1 16 " with regard t
o " }{TEXT 410 5 "dates" }{TEXT -1 117 ". All are agreed on the date (
1844), and even on the approximation theorem (again 1844) that he used
, but not on the " }{TEXT 411 13 "actual number" }{TEXT -1 44 " that h
e first proved to be transcendental. " }}{PARA 0 "" 0 "" {TEXT -1 7 " \+
In " }{TEXT 461 5 "seems" }{TEXT -1 81 " that in his 1844 paper he \+
proved there were transcendental numbers by using his " }{TEXT 466 21
"approximation theorem" }{TEXT -1 5 " and " }{TEXT 462 19 "continued f
ractions" }{TEXT -1 108 " (indeed anyone who knows both can readily cr
eate their own private transcendental numbers at will), but, it" }
{TEXT 465 6 " seems" }{TEXT -1 77 ", it wasn't until 1851 that he gave
the following example (that many authors " }{TEXT 463 5 "state" }
{TEXT -1 44 " he gave in 1844), the number now named the " }{TEXT 467
18 "Liouvillian number" }{TEXT -1 40 ", namely the sum of the infinite
series " }{XPPEDIT 18 0 "Sum(1/(10^m!),m = 1 .. infinity);" "6#-%$Sum
G6$*&\"\"\"F')\"#5-%*factorialG6#%\"mG!\"\"/F-;F'%)infinityG" }{TEXT
-1 24 " ), the decimal number:" }}{PARA 258 "" 0 "" {TEXT -1 2 "0." }
{TEXT 412 2 "11" }{TEXT -1 3 "000" }{TEXT 413 1 "1" }{TEXT -1 17 "0000
0000000000000" }{TEXT 414 1 "1" }{TEXT -1 9 "0000... \n" }}{PARA 0 ""
0 "" {TEXT -1 10 "where the " }{TEXT 350 1 "1" }{TEXT -1 16 "'s occur \+
at the " }{XPPEDIT 18 0 "1^st,2^nd,6^th,24^th,120^th;" "6')\"\"\"%#stG
)\"\"#%#ndG)\"\"'%#thG)\"#CF+)\"$?\"F+" }{TEXT -1 13 ", ... places." }
}{PARA 0 "" 0 "" {TEXT -1 34 " Here is one case (perhaps the " }
{TEXT 363 4 "only" }{TEXT -1 74 " case) where a beginner may follow wi
th little difficulty a transcendence " }{TEXT 355 5 "proof" }{TEXT -1
86 ". It all depends on a completely simple, but utterly important obs
ervation concerning " }{TEXT 360 8 "how well" }{TEXT -1 26 " algebraic
numbers may be " }{TEXT 364 12 "approximated" }{TEXT -1 22 " by ratio
nal numbers: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 31 "Theorem (Liouville, 1844). Let " }{XPPEDIT 18 0 "alpha;"
"6#%&alphaG" }{TEXT -1 40 " be any real algebraic number of degree " }
{TEXT 356 1 "n" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "2 <= n;" "6#1\"\"#%\"
nG" }{TEXT -1 37 "), then there is a positive constant " }{XPPEDIT 18
0 "c = c(alpha);" "6#/%\"cG-F$6#%&alphaG" }{TEXT -1 20 " (i.e. the val
ue of " }{TEXT 357 1 "c" }{TEXT -1 9 " depends " }{TEXT 358 4 "only" }
{TEXT -1 4 " on " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 11 ")
such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%
\"qG!\"\"F," }{TEXT -1 1 " " }{TEXT 359 1 ">" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c(alpha)/(q^n);" "6#*&-%\"cG6#%&alphaG\"\"\")%\"qG%\"nG
!\"\"" }{TEXT -1 10 " ... (i)" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }
{TEXT 365 3 "all" }{TEXT -1 18 " rational numbers " }{XPPEDIT 18 0 "p/
q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 2 " (" }{TEXT 361 1 "p" }
{TEXT -1 5 " and " }{TEXT 362 1 "q" }{TEXT -1 16 " integers, with " }
{TEXT 486 1 "q" }{TEXT -1 7 " > 0). " }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 48 "Remark 1. Liouville's theorem is also t
rue when " }{TEXT 483 2 "n " }{TEXT -1 38 "= 1, in other words in the \+
case where " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 139 " is i
tself a rational number. There is then an obvious modification that ne
eds to be made in the statement of the theorem: instead of \"for " }
{TEXT 484 3 "all" }{TEXT -1 18 " rational numbers " }{XPPEDIT 18 0 "p/
q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 44 " \", one now needs the n
ecessary restriction " }{XPPEDIT 18 0 "p/q <> alpha;" "6#0*&%\"pG\"\"
\"%\"qG!\"\"%&alphaG" }{TEXT -1 65 ". The validity of (i) is then a co
mplete triviality, for suppose " }{XPPEDIT 18 0 "alpha = a/b;" "6#/%&a
lphaG*&%\"aG\"\"\"%\"bG!\"\"" }{TEXT -1 2 " (" }{TEXT 480 1 "a" }
{TEXT -1 5 " and " }{TEXT 481 1 "b" }{TEXT -1 11 " integers, " }{TEXT
485 2 "b " }{TEXT -1 14 "> 0) and that " }{XPPEDIT 18 0 "alpha <> p/q
" "6#0%&alphaG*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 6 ". Then" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha-p/q
);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"F," }{TEXT -1 3 " =
" }{XPPEDIT 18 0 "abs(a/b-p/q);" "6#-%$absG6#,&*&%\"aG\"\"\"%\"bG!\"
\"F)*&%\"pGF)%\"qGF+F+" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs((aq-bp)/
bq) = abs(aq-bp)/bq;" "6#/-%$absG6#*&,&%#aqG\"\"\"%#bpG!\"\"F*%#bqGF,*
&-F%6#,&F)F*F+F,F*F-F," }{TEXT -1 14 ", is at least " }{XPPEDIT 18 0 "
1/bq;" "6#*&\"\"\"F$%#bqG!\"\"" }{TEXT -1 32 ".\nsince the previous nu
merator, " }{XPPEDIT 18 0 "abs(aq-bp)" "6#-%$absG6#,&%#aqG\"\"\"%#bpG!
\"\"" }{TEXT -1 5 ", is " }{TEXT 482 8 "at least" }{TEXT -1 34 " 1 (it
can't be 0 since otherwise " }{XPPEDIT 18 0 "a/b = p/q;" "6#/*&%\"aG
\"\"\"%\"bG!\"\"*&%\"pGF&%\"qGF(" }{TEXT -1 1 ")" }}{PARA 0 "" 0 ""
{TEXT -1 26 "Thus (i) holds with (say) " }{XPPEDIT 18 0 "c = 1/(b+1);
" "6#/%\"cG*&\"\"\"F&,&%\"bGF&F&F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Remark 2. Inequality
(i) is completely trivial if " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"
%\"qG!\"\"" }{TEXT -1 17 " is not close to " }{XPPEDIT 18 0 "alpha;" "
6#%&alphaG" }{TEXT -1 71 ", since, in that case, the left hand side of
(i) is not small, whereas " }{XPPEDIT 18 0 "1/(q^n);" "6#*&\"\"\"F$)%
\"qG%\"nG!\"\"" }{TEXT -1 46 " has minimum value 1, and is quite small
when " }{TEXT 474 1 "q" }{TEXT -1 32 " is large. In short, one should
" }{TEXT 477 10 "appreciate" }{TEXT -1 36 " that the inequality (i) i
s only of " }{TEXT 475 12 "significance" }{TEXT -1 15 " for rationals \+
" }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 10 " tha
t are " }{TEXT 476 5 "close" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "alpha;
" "6#%&alphaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 50 "(The) Proof of Liouville's theorem (is ea
sy). Let " }{XPPEDIT 18 0 "f(x) = a[0]*x^n+a[1]*x^(n-1)+`...`+a[n];" "
6#/-%\"fG6#%\"xG,**&&%\"aG6#\"\"!\"\"\")F'%\"nGF.F.*&&F+6#F.F.)F',&F0F
.F.!\"\"F.F.%$...GF.&F+6#F0F." }{TEXT -1 67 " be the irreducible polyn
omial with integer coefficients such that " }{XPPEDIT 18 0 "f(alpha) =
0;" "6#/-%\"fG6#%&alphaG\"\"!" }{TEXT -1 10 ", and let " }{TEXT 471
1 "p" }{TEXT -1 5 " and " }{TEXT 472 1 "q" }{TEXT -1 18 " be integers \+
with " }{TEXT 473 1 "q" }{TEXT -1 38 " > 0. Then, by the mean value th
eorem," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT
18 0 "f(p/q) = f(p/q)-f(alpha);" "6#/-%\"fG6#*&%\"pG\"\"\"%\"qG!\"\",&
-F%6#*&F(F)F*F+F)-F%6#%&alphaGF+" }{TEXT -1 5 " = ( " }{XPPEDIT 18 0 "
p/q-alpha;" "6#,&*&%\"pG\"\"\"%\"qG!\"\"F&%&alphaGF(" }{TEXT -1 7 " )*
f '(" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 10 ") ... (ii)" }}
{PARA 0 "" 0 "" {TEXT -1 9 "for some " }{XPPEDIT 18 0 "beta;" "6#%%bet
aG" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"q
G!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 12 " (where f '(" }{TEXT 478 1 "x" }{TEXT -1 31 ") is the fir
st derivative of f(" }{TEXT 479 1 "x" }{TEXT -1 4 ")). " }}{PARA 0 ""
0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "1 <= abs(alpha-p/q);" "6#1\"\"\"
-%$absG6#,&%&alphaGF$*&%\"pGF$%\"qG!\"\"F-" }{TEXT -1 40 " then (i) ho
lds (trivially) with (e.g.) " }{XPPEDIT 18 0 "c = 1/2;" "6#/%\"cG*&\"
\"\"F&\"\"#!\"\"" }{TEXT -1 13 ", whereas if " }{XPPEDIT 18 0 "abs(alp
ha-p/q) < 1;" "6#2-%$absG6#,&%&alphaG\"\"\"*&%\"pGF)%\"qG!\"\"F-F)" }
{TEXT -1 7 ", then " }{XPPEDIT 18 0 "abs(beta) < abs(alpha)+1;" "6#2-%
$absG6#%%betaG,&-F%6#%&alphaG\"\"\"F,F," }{TEXT -1 16 ", and thus |f '
(" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 34 ")| is bounded abov
e by a constant " }{TEXT 488 1 "c" }{TEXT -1 56 "' whose value depends
only on (numbers associated with) " }{XPPEDIT 18 0 "alpha;" "6#%&alph
aG" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "
" {TEXT -1 6 " |f '(" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 5 "
)| = " }{XPPEDIT 18 0 "abs(n*a[0]*beta^(n-1)+(n-1)*a[1]*beta^(n-2)+`..
.`+a[n-1]) <= n^2*H*(1+abs(alpha))^(n-1);" "6#1-%$absG6#,**(%\"nG\"\"
\"&%\"aG6#\"\"!F*)%%betaG,&F)F*F*!\"\"F*F**(,&F)F*F*F2F*&F,6#F*F*)F0,&
F)F*\"\"#F2F*F*%$...GF*&F,6#,&F)F*F*F2F**(F)F9%\"HGF*),&F*F*-F%6#%&alp
haGF*,&F)F*F*F2F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 487 2 "H " }{TEXT -1 58 "is
the maximum of the absolute values of the coefficients " }{XPPEDIT
18 0 "a[0],a[1],`...`,a[n-1];" "6&&%\"aG6#\"\"!&F$6#\"\"\"%$...G&F$6#,
&%\"nGF)F)!\"\"" }{TEXT -1 30 ". Multiplying through (ii) by " }
{XPPEDIT 18 0 "q^n;" "6#)%\"qG%\"nG" }{TEXT -1 12 ", and using " }
{XPPEDIT 18 0 "f(p/q) <> 0;" "6#0-%\"fG6#*&%\"pG\"\"\"%\"qG!\"\"\"\"!
" }{TEXT -1 10 " (since f(" }{TEXT 490 1 "x" }{TEXT -1 72 ") is irredu
cible of least degree 2, and so has no rational roots), gives" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "1 <= \+
q^n*abs(f(p/q));" "6#1\"\"\"*&)%\"qG%\"nGF$-%$absG6#-%\"fG6#*&%\"pGF$F
'!\"\"F$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "q^n*abs(alpha-p/q);" "6#*&
)%\"qG%\"nG\"\"\"-%$absG6#,&%&alphaGF'*&%\"pGF'F%!\"\"F/F'" }{TEXT -1
6 "*|f '(" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 5 ")| < " }
{XPPEDIT 18 0 "q^n*abs(alpha-p/q);" "6#*&)%\"qG%\"nG\"\"\"-%$absG6#,&%
&alphaGF'*&%\"pGF'F%!\"\"F/F'" }{TEXT -1 1 "*" }{TEXT 489 1 "c" }
{TEXT -1 1 "'" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 24 "(i) follows immediately." }}{PARA 258 "" 0 "" {TEXT -1
19 "___________________" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 47 "I include the following only to illustrate the " }
{TEXT 464 4 "idea" }{TEXT -1 64 " behind the proof of Liouville's theo
rem. I choose an algebraic " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 67 " of degree 3, one that is a solution of the (irreducible)
equation " }{XPPEDIT 18 0 "6*x^3-14*x-x+11 = 0;" "6#/,**&\"\"'\"\"\"*
$%\"xG\"\"$F'F'*&\"#9F'F)F'!\"\"F)F-\"#6F'\"\"!" }{TEXT -1 1 ":" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "f := x -> 6*x^3 - 14*x^2 - x + 11;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(f(x));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f(x), x = -1..2);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 9 "I choose " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 57 " t
o be the largest of those solutions (you can see it is " }{TEXT 491
13 "slightly more" }{TEXT -1 15 " than 1.925...)" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 17 "fsolve(f(x) = 0);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and a ratio
nal number " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT
-1 9 " that is " }{TEXT 470 4 "near" }{TEXT -1 4 " to " }{XPPEDIT 18
0 "alpha;" "6#%&alphaG" }{TEXT -1 41 " (in the following diagram I hav
e chosen " }{TEXT 495 3 "two" }{TEXT -1 36 " rational numbers that are
close to " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "p1/q1 = 19/10;" "6#/*&%#p1G\"\"\"%#q1G!\"\"*&\"#>F&\"#5
F(" }{TEXT -1 26 ", slightly to the left of " }{XPPEDIT 18 0 "alpha;"
"6#%&alphaG" }{TEXT -1 5 ", and" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "p2/
q2 = 39/20;" "6#/*&%#p2G\"\"\"%#q2G!\"\"*&\"#RF&\"#?F(" }{TEXT -1 27 "
, slightly to the right of " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 15 "Then, close in " }{TEXT 493 4 "near" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 31 ", the graph of the f
unction is " }{TEXT 492 13 "almost linear" }{TEXT -1 20 ", and thus (w
hether " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1
4 " be " }{XPPEDIT 18 0 "p1/q1;" "6#*&%#p1G\"\"\"%#q1G!\"\"" }{TEXT
-1 4 " or " }{XPPEDIT 18 0 "p2/q2;" "6#*&%#p2G\"\"\"%#q2G!\"\"" }
{TEXT -1 3 " ):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(f(p/q));" "6#-%$absG6#-%\"fG6#*&%\"
pG\"\"\"%\"qG!\"\"" }{TEXT -1 57 " (which is a non-zero rational numbe
r - with denominator " }{XPPEDIT 18 0 "q^3;" "6#*$%\"qG\"\"$" }{TEXT
-1 28 " - and so has minimum value " }{XPPEDIT 18 0 "1/(q^3);" "6#*&\"
\"\"F$*$%\"qG\"\"$!\"\"" }{TEXT -1 3 " ) " }}{PARA 258 "" 0 "" {TEXT
-1 3 "is " }{TEXT 494 6 "almost" }{TEXT -1 10 " equal to " }{XPPEDIT
18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"
F," }{TEXT -1 6 "*|f '(" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT
-1 3 ")| " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
31 "It should be clear, then, that " }{XPPEDIT 18 0 "abs(alpha-p/q)" "
6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"F," }{TEXT -1 13 " is bo
unded, " }{TEXT 496 10 "from below" }{TEXT -1 30 ", by a constant mult
iplied by " }{XPPEDIT 18 0 "1/(q^3);" "6#*&\"\"\"F$*$%\"qG\"\"$!\"\""
}{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "with(plo
ts): with(plottools):\npl1 := plot(f(x), x = 1.89..1.955, thickness=2)
:\npl2 := textplot([1.93,-0.03,`alpha`],align=LEFT):\np1 := 19: q1 := \+
10:\nl1 := line([p1/q1, 0], [p1/q1, f(p1/q1)], color=navy, thickness=2
):\np2 := 39: q2 := 20:\nl2 := line([p2/q2, 0], [p2/q2, f(p2/q2)], col
or=brown, thickness=2):\npl3 := textplot([1.9,0.06,`p1/q1`],align=ABOV
E):\npl4 := textplot([1.9,-0.3,`f(p1/q1)`],align=RIGHT):\npl5 := textp
lot([1.95,-0.02,`p2/q2`],align=BELOW):\npl6 := textplot([1.95,0.35,`f(
p2/q2)`],align=LEFT):\ndisplay([pl1, pl2, l1, l2, pl3, pl4, pl5, pl6])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 376 "Once one is in possessio
n of Liouville's theorem, transcendental numbers simply fall into ones
lap (in fact, with Cantorian hindsight, one may construct an uncounta
ble number of such Liouville-type numbers). Anyone who is familiar wit
h continued fraction expansions of irrational numbers will see immedia
tely that all one has to do - to produce transcendental numbers - is t
o " }{TEXT 497 6 "define" }{TEXT -1 335 " numbers whose partial quotie
nts grow in size sufficiently rapidly... I think that is what Liouvill
e did in his 1844 paper (I've never read the original paper, and stand
to be corrected), whereas it was in a later (1851) paper that he gave
(what many seem to think was in the 1844 one) his well-known decimal \+
example mentioned earlier:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "alpha = Sum(1/(10^m!),m = 1 \+
.. infinity);" "6#/%&alphaG-%$SumG6$*&\"\"\"F))\"#5-%*factorialG6#%\"m
G!\"\"/F/;F)%)infinityG" }{TEXT -1 6 " = 0." }{TEXT 498 2 "11" }
{TEXT -1 3 "000" }{TEXT 499 1 "1" }{TEXT -1 17 "00000000000000000" }
{TEXT 500 1 "1" }{TEXT -1 9 "0000... " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 26 "Proof of transcendence of " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 18 " (the real number "
}{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 " " }{TEXT 503 6 "can
not" }{TEXT -1 66 " be algebraic since it possesses rational approxima
tions that are " }{TEXT 502 12 "incompatible" }{TEXT -1 92 " with it b
eing algebraic; the relevant rational approximations are simply the ea
rly, (first " }{TEXT 504 1 "m" }{TEXT -1 45 " terms, varying part), of
the infinite sum). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 ""
0 "" {XPPEDIT 18 0 "Sum(1/(10^m!),m = 1 .. infinity) = 1/(10^1!)+1/(10
^2!)+`...`+1/(10^m!)+1/(10^(m+1)!)+`...`;" "6#/-%$SumG6$*&\"\"\"F()\"#
5-%*factorialG6#%\"mG!\"\"/F.;F(%)infinityG,.*&F(F()F*-F,6#F(F/F(*&F(F
()F*-F,6#\"\"#F/F(%$...GF(*&F(F()F*-F,6#F.F/F(*&F(F()F*-F,6#,&F.F(F(F(
F/F(F=F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "1/(10^1!)+1/(10^2!)+`...`+1/(1
0^m!);" "6#,**&\"\"\"F%)\"#5-%*factorialG6#F%!\"\"F%*&F%F%)F'-F)6#\"\"
#F+F%%$...GF%*&F%F%)F'-F)6#%\"mGF+F%" }{TEXT -1 22 " is a rational num
ber " }{XPPEDIT 18 0 "p[m]/q[m];" "6#*&&%\"pG6#%\"mG\"\"\"&%\"qG6#F'!
\"\"" }{TEXT -1 18 " with denominator " }{XPPEDIT 18 0 "q[m] = 10^m!;
" "6#/&%\"qG6#%\"mG)\"#5-%*factorialG6#F'" }{TEXT -1 10 ", and thus" }
}{PARA 0 "" 0 "" {XPPEDIT 18 0 "alpha = p[m]/q[m]+R[m];" "6#/%&alphaG,
&*&&%\"pG6#%\"mG\"\"\"&%\"qG6#F*!\"\"F+&%\"RG6#F*F+" }{TEXT -1 28 ", w
here the remainder term, " }{XPPEDIT 18 0 "R[m];" "6#&%\"RG6#%\"mG" }
{TEXT -1 5 ", is " }{TEXT 505 8 "positive" }{TEXT -1 5 " and " }{TEXT
501 4 "less" }{TEXT -1 6 " than " }{XPPEDIT 18 0 "2/(10^(m+1)!) = 2/(q
[m]^(m+1));" "6#/*&\"\"#\"\"\")\"#5-%*factorialG6#,&%\"mGF&F&F&!\"\"*&
F%F&)&%\"qG6#F-,&F-F&F&F&F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha-p/q) < 2/(q[m]^(m
+1));" "6#2-%$absG6#,&%&alphaG\"\"\"*&%\"pGF)%\"qG!\"\"F-*&\"\"#F))&F,
6#%\"mG,&F3F)F)F)F-" }{TEXT -1 15 " (true for all " }{TEXT 506 2 "m "
}{TEXT -1 19 "= 1, 2, 3, 4, ... )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 44 "That latter inequality is incompatible wi
th " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 35 " being an alge
braic number: for if " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1
26 " were algebraic of degree " }{TEXT 507 2 "n " }{TEXT -1 20 "then o
ne would have " }{XPPEDIT 18 0 "2/(q[m]^(m+1));" "6#*&\"\"#\"\"\")&%\"
qG6#%\"mG,&F*F%F%F%!\"\"" }{TEXT -1 1 " " }{TEXT 508 1 ">" }{TEXT -1
1 " " }{XPPEDIT 18 0 "c(alpha)/(q[m]^n);" "6#*&-%\"cG6#%&alphaG\"\"\")
&%\"qG6#%\"mG%\"nG!\"\"" }{TEXT -1 34 ", which is clearly impossible f
or " }{TEXT 509 1 "m" }{TEXT -1 20 " sufficiently large." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "Cantor motivated \+
observation. With Cantorian hindsight one may in fact observe that the
re are an uncountable number of transcendental numbers of Liouville ty
pe (meaning ones whose transcendence may be established by the Liouvil
le approach). Simply let \{" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG"
}{TEXT -1 36 "\} be any infinite sequence in which " }{XPPEDIT 18 0 "a
[n] = 1;" "6#/&%\"aG6#%\"nG\"\"\"" }{TEXT -1 14 " or 2 for all " }
{TEXT 1227 1 "n" }{TEXT -1 100 ". There is an uncountable number of su
ch sequences, and, for any one of those sequences, the number " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 12 " defined by " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "alpha = Sum(a[m]/(10^m!),m = 1 .. infinity);" "6#/%&alp
haG-%$SumG6$*&&%\"aG6#%\"mG\"\"\")\"#5-%*factorialG6#F,!\"\"/F,;F-%)in
finityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 101 "is transcendental, and no two such numbers are equa
l (different sequences produce different numbers, " }{TEXT 1228 12 "in
this case" }{TEXT -1 2 ")." }}{PARA 258 "" 0 "" {TEXT -1 16 "________
________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
193 "Actually, there is a way of seeing that the above number is trans
cendental, without knowing anything about the above approximation theo
rem. Indeed this alternative way could be understood by a " }{TEXT
510 21 "numerate school pupil" }{TEXT -1 65 " with some ability. All t
hat is needed is that one should know - " }{TEXT 468 12 "really know!
" }{TEXT -1 63 " - how to multiply decimals together, as I will now il
lustrate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
514 8 "Question" }{TEXT -1 43 ". Which do you think is easier to work \+
out:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 4 "t
he " }{TEXT 511 6 "square" }{TEXT -1 32 " of the infinite decimal numb
er " }{TEXT 469 10 "0.01010101" }{TEXT -1 9 "... ?, or" }}{PARA 15 ""
0 "" {TEXT -1 4 "the " }{TEXT 513 6 "square" }{TEXT -1 42 " of the inf
inite Liouville decimal number " }{TEXT 512 10 "0.11000100" }{TEXT -1
5 "... ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT
-1 188 "If you tried to do the first one longhand (as we all learned a
t school) I think you would soon run into trouble. Try it! Ask your pu
pils to try it. The 'carrying' gets to be troublesome..." }}{PARA 259
"" 0 "" {TEXT -1 71 " It shouldn't surprise you to know that the sq
uared decimal will be " }{TEXT 515 8 "periodic" }{TEXT -1 83 ": that's
simply because the above decimal - which should be seen as the sum of
the " }{TEXT 516 25 "infinite geometric series" }{TEXT -1 19 " with i
nitial term " }{XPPEDIT 18 0 "1/100;" "6#*&\"\"\"F$\"$+\"!\"\"" }
{TEXT -1 20 " and 'common ratio' " }{XPPEDIT 18 0 "1/100;" "6#*&\"\"\"
F$\"$+\"!\"\"" }{TEXT -1 63 " - is, of course, the decimal expansion o
f the rational number " }{XPPEDIT 18 0 "1/99;" "6#*&\"\"\"F$\"#**!\"\"
" }{TEXT -1 85 ", and thus the sought, squared decimal, will be whatev
er is the decimal expansion of " }{XPPEDIT 18 0 "1/(99^2);" "6#*&\"\"
\"F$*$\"#**\"\"#!\"\"" }{TEXT -1 116 " . But what are the digits in t
hat expansion? First, here are the already seen digits in the infinite
expansion of " }{XPPEDIT 18 0 "1/99;" "6#*&\"\"\"F$\"#**!\"\"" }
{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numthe
ory):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pdexpand(1/99);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 87 "That final \"[0, 1]\" is to be repeated ad infinitum to
produce the decimal expansion of " }{XPPEDIT 18 0 "1/99;" "6#*&\"\"\"
F$\"#**!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "Here are \+
a few others before we see the decimal expansion of " }{XPPEDIT 18 0 "
1/(99^2);" "6#*&\"\"\"F$*$\"#**\"\"#!\"\"" }{TEXT -1 2 " :" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pdexpand(1/7);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 17 "which means that " }{XPPEDIT 18 0 "1/7 = .142857*`
...ad inf`;" "6#/*&\"\"\"F%\"\"(!\"\"*&$\"'dG9!\"'F%%*...ad~infGF%" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pdexpand(1/28);" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 17 "which means that " }{XPPEDIT 18 0 "1/28 =
.3571428e-1*`...ad inf`;" "6#/*&\"\"\"F%\"#G!\"\"*&$\"(G9d$!\")F%%*..
.ad~infGF%" }{TEXT -1 41 " (with only the '571428' being repeated) " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "pdexpand(-97/26);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "which means that " }{XPPEDIT 18 0
"(-97)/26 = -3.7307692*`...ad inf`;" "6#/*&,$\"#(*!\"\"\"\"\"\"#EF',$*
&$\")#p2t$!\"(F(%*...ad~infGF(F'" }{TEXT -1 41 " (with only the '30769
2' being repeated) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 38 "And finally, the decimal expansion of " }{XPPEDIT 18 0
"(.1010101e-1*`...`)^2;" "6#*$*&$\"(,,,\"!\")\"\"\"%$...GF(\"\"#" }
{TEXT -1 47 " is (anyone like to guess before I reveal it?):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "pdexpand(1/99^2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 117 "There are 198 digits in the repeating cycle. Don't try -
by hand! - working out the decimal expansion of the cube of " }
{XPPEDIT 18 0 "1/99;" "6#*&\"\"\"F$\"#**!\"\"" }{TEXT -1 127 ": there \+
are 19602 digits in the cycle. Anyone familiar with the relevant Numbe
r Theory will know it's to do with the fact that " }{XPPEDIT 18 0 "ord
[10](99^3);" "6#-&%$ordG6#\"#56#*$\"#**\"\"$" }{TEXT -1 14 " = 19602 (
= 2*" }{XPPEDIT 18 0 "99^2;" "6#*$\"#**\"\"#" }{TEXT -1 63 "), and the
above decimal expansion is to do with the fact that " }{XPPEDIT 18 0
"ord[10](99^2);" "6#-&%$ordG6#\"#56#*$\"#**\"\"#" }{TEXT -1 16 " = 198
(= 2*99):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "order(10, 99^
2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 61 "and thus that the periodic block of the decimal ex
pansion of " }{XPPEDIT 18 0 "1/(99^2);" "6#*&\"\"\"F$*$\"#**\"\"#!\"\"
" }{TEXT -1 38 " commences with three 0s, followed by:" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "(10^198 - 1)/99^2;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 442 "It might initially surprise the unsuspecting that it's much ea
sier to work out - by hand! - the decimal value of the square of the a
bove Liouvillian decimal. And its cube, and its fourth power, and ... \+
. Of course one sees why it's easier... : those huge blocks of 0's bet
ween the 1's. That alone - without having to use Liouville's approxima
tion theorem - allows one to argue that the Louville decimal is transc
endental, as I now illustrate." }}{PARA 0 "" 0 "" {TEXT -1 11 " Fir
st I" }{TEXT 775 12 " manufacture" }{TEXT -1 33 " a certain quadratic \+
polynomial, " }{XPPEDIT 18 0 "50*x^2+4949*x-545;" "6#,(*&\"#]\"\"\"*$%
\"xG\"\"#F&F&*&\"%\\\\F&F(F&F&\"$X&!\"\"" }{TEXT -1 40 ", for which th
e above Liouville number (" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 5 ") is " }{TEXT 774 6 "almost" }{TEXT -1 14 " a root (i.e. \+
" }{XPPEDIT 18 0 "x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1 4 " is " }
{TEXT 773 6 "almost" }{TEXT -1 15 " a solution of " }{XPPEDIT 18 0 "50
*x^2+4949*x-545 = 0;" "6#/,(*&\"#]\"\"\"*$%\"xG\"\"#F'F'*&\"%\\\\F'F)F
'F'\"$X&!\"\"\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 12 " \+
This is " }{TEXT 776 3 "how" }{TEXT -1 242 " I arrived at that quadr
atic (don't worry if you don't follow the continued fraction reasoning
behind it): I began by forming the rational number corresponding to t
he initial part of the Liouville number; in fact simply the third part
ial sum " }{XPPEDIT 18 0 "1/(10^1!)+1/(10^2!)+1/(10^3!);" "6#,(*&\"\"
\"F%)\"#5-%*factorialG6#F%!\"\"F%*&F%F%)F'-F)6#\"\"#F+F%*&F%F%)F'-F)6#
\"\"$F+F%" }{TEXT -1 85 " , which is a rational number, and so has a t
erminating continued fraction expansion:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 37 "alpha3 := add(1/(10)^(n!), n = 1..3);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cfrac(alpha3, quotients);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 286 "Anyone who knows about continued fractions, and knows th
e effect of a large partial quotient, will spot that '99', and realise
why I now define the infinite periodic continued fraction with partia
l quotients [0, 9, 11, 99, 9, 11, 99, ... ], having initial 0, followe
d by the infinitely " }{TEXT 460 8 "repeated" }{TEXT -1 19 " block [9,
11, 99]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "cf3 := [[0], [
9, 11, 99]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "beta3 := in
vcfrac(cf3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Digits := 1
0:\nevalf(beta3); # you see the 'extra' bit:" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "That num
ber, beta3 (which is close to alpha3, which is close to " }{XPPEDIT
18 0 "alpha;" "6#%&alphaG" }{TEXT -1 85 "), is the solution of a quadr
atic equation, and that equation is recovered by setting" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0
"x = -4949/100+sqrt(24601601)/100;" "6#/%\"xG,&*&\"%\\\\\"\"\"\"$+\"!
\"\"F**&-%%sqrtG6#\"),;gCF(F)F*F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "10
0*x+4949 = sqrt(24601601);" "6#/,&*&\"$+\"\"\"\"%\"xGF'F'\"%\\\\F'-%%s
qrtG6#\"),;gC" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "(100*x+4949)^2 = 2460
1601;" "6#/*$,&*&\"$+\"\"\"\"%\"xGF(F(\"%\\\\F(\"\"#\"),;gC" }{TEXT
-1 6 ", etc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(expand((10
0*x + 4949)^2) - 24601601);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Th
e coefficients obviously have gcd = 200:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 28 "igcd(10000, 989800, 109000);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 51 "solve(50*x^2 + 4949*x - 545 = 0); # the first is
L3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 22 "Now observe how close " }{TEXT 778 1 "x" }{TEXT
-1 49 " = alpha3 is to being a solution of the equation " }{XPPEDIT
18 0 "50*x^2+4949*x-545 = 0" "6#/,(*&\"#]\"\"\"*$%\"xG\"\"#F'F'*&\"%\\
\\F'F)F'F'\"$X&!\"\"\"\"!" }{TEXT -1 32 ". I will calculate the value \+
of\n" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "50*x^2+4949*x;
" "6#,&*&\"#]\"\"\"*$%\"xG\"\"#F&F&*&\"%\\\\F&F(F&F&" }{TEXT -1 5 " at
" }{TEXT 777 1 "x" }{TEXT -1 46 " = alpha3, first to 7 decimal place
s, and then" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "50*x^2+4949*x;" "6#,&*&
\"#]\"\"\"*$%\"xG\"\"#F&F&*&\"%\\\\F&F(F&F&" }{TEXT -1 5 " at " }
{TEXT 800 1 "x" }{TEXT -1 31 " = alpha3, to 8 decimal places\n" }}
{PARA 0 "" 0 "" {TEXT -1 12 "and observe " }{TEXT 799 16 "proximity to
545" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "eva
lf(50*alpha3^2 + 4949*alpha3, 7); \nevalf(50*alpha3^2 + 4949*alpha3, 8
); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 5 "Thus " }{TEXT 779 1 "x" }{TEXT -1 13 " = alpha3 \+
is " }{TEXT 801 6 "almost" }{TEXT -1 28 " a solution of the equation \+
" }{XPPEDIT 18 0 "50*x^2+4949*x-545 = 0;" "6#/,(*&\"#]\"\"\"*$%\"xG\"
\"#F'F'*&\"%\\\\F'F)F'F'\"$X&!\"\"\"\"!" }{TEXT -1 26 ", and we reason
ably ask: \n" }}{PARA 258 "" 0 "" {TEXT -1 6 "could " }{XPPEDIT 18 0 "
x = alpha;" "6#/%\"xG%&alphaG" }{TEXT -1 1 " " }{TEXT 780 2 "be" }
{TEXT -1 12 " a solution?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 98 "Without using Liouville's approximation theorem we
may easily see that it isn't. I'm not going to " }{TEXT 802 31 "'dot \+
every i and cross every t'" }{TEXT -1 64 ", but rather give pointers a
nd invite you to ponder on your own." }}{PARA 0 "" 0 "" {TEXT -1 45 "W
hat one wants to do is to somehow calculate " }{XPPEDIT 18 0 "50*alpha
^2+4949*alpha-545;" "6#,(*&\"#]\"\"\"*$%&alphaG\"\"#F&F&*&\"%\\\\F&F(F
&F&\"$X&!\"\"" }{TEXT -1 5 " and " }{TEXT 803 3 "see" }{TEXT -1 17 " t
hat it isn't 0." }}{PARA 0 "" 0 "" {TEXT -1 43 "Note first of all that
Maple can't do that " }{TEXT 808 5 "exact" }{TEXT -1 31 " calculation
for us (though it " }{TEXT 804 3 "can" }{TEXT -1 24 " do many infinit
e sums):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "restart;\nalpha
:= sum(1/10^(n!), n = 1..infinity); # 'add' won't 'work'" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "50*alpha^2 + 4949*alpha - 545;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 33 "But one can calculate, or rather " }{TEXT 805 3 "see" }
{TEXT -1 5 " the " }{TEXT 806 9 "essential" }{TEXT -1 34 " part of the
decimal expansion of " }{XPPEDIT 18 0 "50*alpha^2+4949*alpha-545" "6#
,(*&\"#]\"\"\"*$%&alphaG\"\"#F&F&*&\"%\\\\F&F(F&F&\"$X&!\"\"" }{TEXT
-1 29 " by hand, and in the process " }{TEXT 807 3 "see" }{TEXT -1 6 "
that " }{XPPEDIT 18 0 "50*alpha^2+4949*alpha-545 <> 0;" "6#0,(*&\"#]
\"\"\"*$%&alphaG\"\"#F'F'*&\"%\\\\F'F)F'F'\"$X&!\"\"\"\"!" }{TEXT -1
65 ". The essential point is that it's actually quite easy to square \+
" }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 3 " - " }{TEXT 811
11 "much easier" }{TEXT -1 88 " than squaring the earlier challenge de
cimal .0101010101... - in the sense that one can " }{TEXT 810 3 "see"
}{TEXT -1 28 " what its decimal expansion " }{TEXT 809 2 "is" }{TEXT
-1 243 " (this is difficult to type up, since I want vertical holds on
the place values in the good old-fashioned way of multiplying and 'ca
rrying'). For each line you should keep in mind where the next '1' is \+
occurs, and the next, and the next, ... :" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 " .1100010000
0000000000000100000000... multiplied by" }}{PARA 0 "" 0 "" {TEXT -1
62 " .11000100000000000000000100000000... is" }}
{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 65 " \+
.011000100000000000000000100000000... plus" }}{PARA 0
"" 0 "" {TEXT -1 66 " .0011000100000000000000000
100000000... plus" }}{PARA 0 "" 0 "" {TEXT -1 70 " \+
.00000011000100000000000000000100000000... plus" }}{PARA 0 "" 0 ""
{TEXT -1 71 " .000000000000000000000000110001000
00000000000000" }{TEXT 813 1 "1" }{TEXT -1 16 "00000000... plus" }}
{PARA 0 "" 0 "" {TEXT -1 24 " " }{TEXT 812 12 "
ad infinitum" }{TEXT -1 18 ", which adds up to" }}{PARA 0 "" 0 ""
{TEXT -1 13 " " }{XPPEDIT 18 0 "alpha^2;" "6#*$%&alphaG\"
\"#" }{TEXT -1 52 " = .0121002200010000000000002200020000000000000000
0" }{TEXT 814 1 "1" }{TEXT -1 18 "00000000... , thus" }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "50*alpha^2;" "6#*&\"#]\"\"\"*$%&alpha
G\"\"#F%" }{TEXT -1 58 " = .6050110000500000000000110001000000
000000000000" }{TEXT 815 2 "50" }{TEXT -1 10 "0000000..." }}{PARA 0 "
" 0 "" {XPPEDIT 18 0 "4949*alpha;" "6#*&\"%\\\\\"\"\"%&alphaGF%" }
{TEXT -1 82 " = 544.3949490000000000000049490000000000000000000000000
000... , which adds up to" }}{PARA 0 "" 0 "" {TEXT -1 67 " \+
544.9999600000500000000049600001000000000000000000" }{TEXT 816 2
"50" }{TEXT -1 8 "0000... " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 82 "and what I am attempting to draw your eye's atten
tion to is the occurence of that " }{TEXT 817 2 "50" }{TEXT 818 0 "" }
{TEXT -1 92 ", which is going to occur over and over again (as a conse
quence of which it is obvious that " }{XPPEDIT 18 0 "50*alpha^2+4949*a
lpha;" "6#,&*&\"#]\"\"\"*$%&alphaG\"\"#F&F&*&\"%\\\\F&F(F&F&" }{TEXT
-1 1 " " }{TEXT 821 6 "cannot" }{TEXT -1 63 " be equal to 545, a whole
number). I invite you to think about " }{TEXT 819 3 "why" }{TEXT -1
41 " that kind of thing happens, not just in " }{TEXT 820 4 "this" }
{TEXT -1 176 " particular case but in the general case. The above kind
of numerical play is only to give one a feeling for this, but you nee
d to start thinking about the question: where, in " }{TEXT 822 5 "plac
e" }{TEXT -1 63 ", do we get 1s occuring in the decimal expansion of (
not just) " }{XPPEDIT 18 0 "alpha^2;" "6#*$%&alphaG\"\"#" }{TEXT -1 5
"(but " }{XPPEDIT 18 0 "alpha^3,alpha^4,`...`;" "6%*$%&alphaG\"\"$*$F$
\"\"%%$...G" }{TEXT -1 131 " ), and what then if the effect of multipl
ying by coefficients, and adding, ... Can we end up getting '0' after \+
it's all totted up?" }}{PARA 0 "" 0 "" {TEXT -1 120 " The numerical
play has a more formal aspect to it (which should be preceded by play
), and that formal aspect is the " }{TEXT 823 8 "infinite" }{TEXT -1
19 " version of (e.g.) " }{XPPEDIT 18 0 "(a[1]+a[2]+a[3])^2 = a[1]^2+a
[2]^2+a[3]^2+2*a[1]*a[2]+2*a[1]*a[2]+2*a[2]*a[3];" "6#/*$,(&%\"aG6#\"
\"\"F)&F'6#\"\"#F)&F'6#\"\"$F)F,,.*$&F'6#F)F,F)*$&F'6#F,F,F)*$&F'6#F/F
,F)*(F,F)&F'6#F)F)&F'6#F,F)F)*(F,F)&F'6#F)F)&F'6#F,F)F)*(F,F)&F'6#F,F)
&F'6#F/F)F)" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 7 "namely " }
{XPPEDIT 18 0 "(a[1]+a[2]+a[3]+`...`)^2 = a[1]^2+a[2]^2+a[3]^2+`...`+2
*a[1]*a[2]+2*a[1]*a[3]+2*a[1]*a[4]+`...`;" "6#/*$,*&%\"aG6#\"\"\"F)&F'
6#\"\"#F)&F'6#\"\"$F)%$...GF)F,,2*$&F'6#F)F,F)*$&F'6#F,F,F)*$&F'6#F/F,
F)F0F)*(F,F)&F'6#F)F)&F'6#F,F)F)*(F,F)&F'6#F)F)&F'6#F/F)F)*(F,F)&F'6#F
)F)&F'6#\"\"%F)F)F0F)" }}{PARA 0 "" 0 "" {TEXT -1 2 "+ " }{XPPEDIT 18
0 "2*a[2]*a[3]+2*a[2]*a[4]+2*a[2]*a[5]+`...`;" "6#,**(\"\"#\"\"\"&%\"a
G6#F%F&&F(6#\"\"$F&F&*(F%F&&F(6#F%F&&F(6#\"\"%F&F&*(F%F&&F(6#F%F&&F(6#
\"\"&F&F&%$...GF&" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*a[3]*a[4]+2*a[3
]*a[5]+2*a[3]*a[6]+`...`;" "6#,**(\"\"#\"\"\"&%\"aG6#\"\"$F&&F(6#\"\"%
F&F&*(F%F&&F(6#F*F&&F(6#\"\"&F&F&*(F%F&&F(6#F*F&&F(6#\"\"'F&F&%$...GF&
" }{TEXT -1 3 " + " }{TEXT 827 12 "ad infinitum" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "so as to see where 1s occ
ur in " }{XPPEDIT 18 0 "(.110001000*`...`)^2 = `.1`^2+`.01`^2+`.000001
`^2+`...`;" "6#/*$*&$\"*+5+5\"!\"*\"\"\"%$...GF)\"\"#,**$%#.1GF+F)*$%$
.01GF+F)*$%(.000001GF+F)F*F)" }{TEXT -1 31 " (each of those contribute
s a '" }{TEXT 828 1 "1" }{TEXT -1 78 "' to the decimal expansion) + th
e 'cross terms', each of which contributes a '" }{TEXT 829 1 "2" }
{TEXT -1 48 "' to the decimal, and to see how isolated those " }{TEXT
830 1 "1" }{TEXT -1 24 "s are in terms of their " }{TEXT 824 21 "neigh
bouring non-zero" }{TEXT -1 9 " digits. " }}{PARA 0 "" 0 "" {TEXT -1
57 " I will end with a lovely example from Conway & Guy's " }{TEXT
826 19 "The Book of Numbers" }{TEXT -1 44 ". They observe that the Lio
uville number is " }{TEXT 825 6 "almost" }{TEXT -1 50 " a solution of \+
the 6th degree polynomial equation " }{XPPEDIT 18 0 "10*x^6-75*x^3-190
*x+21 = 0" "6#/,**&\"#5\"\"\"*$%\"xG\"\"'F'F'*&\"#vF'*$F)\"\"$F'!\"\"*
&\"$!>F'F)F'F/\"#@F'\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 86 "Digits := 24: alp := evalf(add(1/(10)^(n!), n = 1..
4)):\n10*alp^6 - 75*alp^3 - 190*alp;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 107 "They don't tell their readers how that polynomial was ar
rived at, and I leave it to my reader to speculate." }}}{SECT 1 {PARA
4 "" 0 "" {TEXT -1 51 "Fast track observations & questions after Liouv
ille" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "O
bservation. In connection with the irrationality of (e.g.) " }
{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 63 " one comes \+
to know that the important result that the equation " }{XPPEDIT 18 0 "
x^2-2*y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(!\"\"F(" }
{TEXT -1 60 " has an infinite number of solutions in (positive) intege
rs " }{TEXT 1083 1 "x" }{TEXT -1 2 ", " }{TEXT 1218 1 "y" }{TEXT -1
56 ". One may think of that as starting from the meaning of " }
{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 51 "'s irration
ality: there are no (positive) integers " }{TEXT 1219 4 "x, y" }{TEXT
-1 11 " such that " }{XPPEDIT 18 0 "x^2 = 2*y^2;" "6#/*$%\"xG\"\"#*&F&
\"\"\"*$%\"yGF&F(" }{TEXT -1 13 ", and so the " }{TEXT 1220 9 "next be
st" }{TEXT -1 18 " thing is to have " }{XPPEDIT 18 0 "x^2 = 2*y^2+1;"
"6#/*$%\"xG\"\"#,&*&F&\"\"\"*$%\"yGF&F)F)F)F)" }{TEXT -1 5 " (or " }
{XPPEDIT 18 0 "x^2 = 2*y^2-1;" "6#/*$%\"xG\"\"#,&*&F&\"\"\"*$%\"yGF&F)
F)F)!\"\"" }{TEXT -1 125 ") (An interested reader, not familiar with t
his, would benefit, I believe, from reading my web site notes on - wha
t I call - " }{TEXT 1221 25 "L- and R-approximations. " }{TEXT -1 25 "
In general it's not just " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"
#" }{TEXT -1 32 " that would be of interest, but " }{XPPEDIT 18 0 "sqr
t(d);" "6#-%%sqrtG6#%\"dG" }{TEXT -1 20 ", and thus not just " }
{XPPEDIT 18 0 "x^2-2*y^2 = 1" "6#/,&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F
(!\"\"F(" }{TEXT -1 6 ", but " }{XPPEDIT 18 0 "x^2-d*y^2 = 1;" "6#/,&*
$%\"xG\"\"#\"\"\"*&%\"dGF(*$%\"yGF'F(!\"\"F(" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 15 " Every such " }{TEXT 1222 1 "x" }
{TEXT -1 2 ", " }{TEXT 1223 1 "y" }{TEXT -1 45 " leads to a fantastic \+
rational approximation " }{XPPEDIT 18 0 "x/y;" "6#*&%\"xG\"\"\"%\"yG!
\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#
" }{TEXT -1 132 ", as good an approximation, in fact, as there can pos
sibly be. Slightly throwing away some of the quality of approximation,
one has:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "abs(sqrt(2)-x/y);" "6#-%$absG6#,&-%%sqrtG6#\"\"#\"\"\"*
&%\"xGF+%\"yG!\"\"F/" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "1/(y^2);" "6#*
&\"\"\"F$*$%\"yG\"\"#!\"\"" }{TEXT -1 34 " (infinitely often, in fact)
(i)" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT
-1 91 " Now, inequality (i) happens to be (independently) gaurantee
d by the completely general:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 25 "Theorem (Dirichlet). Let " }{XPPEDIT 18 0 "alph
a;" "6#%&alphaG" }{TEXT -1 62 " be any real irrational number (and it \+
doesn't matter whether " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT
-1 88 " is algebraic or transcendental), then there are an infinite nu
mber of rational numbers " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"qG
!\"\"" }{TEXT -1 2 " (" }{TEXT 1224 1 "p" }{TEXT -1 2 ", " }{TEXT
1225 1 "q" }{TEXT -1 11 " integers, " }{TEXT 1226 1 "q" }{TEXT -1 15 "
> 0) such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%
\"qG!\"\"F," }{TEXT -1 3 " < " }{XPPEDIT 18 0 "1/(q^2);" "6#*&\"\"\"F$
*$%\"qG\"\"#!\"\"" }{TEXT -1 12 " ... (ii)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Question. Are there any s
uch " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 58 "s for which a
better approximation than (ii) could happen?" }}{PARA 0 "" 0 ""
{TEXT -1 131 "Answer. Yes, of course. It's easy. By simply varying the
type of number encountered in the Liouville section, and forming a nu
mber " }{TEXT 1229 4 "like" }{TEXT -1 8 " (e.g.) " }{XPPEDIT 18 0 "alp
ha = Sum(1/(10^(3^m)),m = 1 .. infinity);" "6#/%&alphaG-%$SumG6$*&\"\"
\"F))\"#5)\"\"$%\"mG!\"\"/F.;F)%)infinityG" }{TEXT -1 21 " then one ob
tains an " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 25 " for whi
ch the inequality" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "
" {XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF
(%\"qG!\"\"F," }{TEXT -1 3 " < " }{XPPEDIT 18 0 "1/(q^3);" "6#*&\"\"\"
F$*$%\"qG\"\"$!\"\"" }{TEXT -1 13 " ... (iii)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "has an infinite number of
solutions in rational numbers " }{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"
\"%\"qG!\"\"" }{TEXT -1 98 " (a careful reader will immediately spot t
hat I haven't quite got the full validity of (iii), but " }{TEXT 1230
6 "almost" }{TEXT -1 5 ", and" }{TEXT 1232 1 " " }{TEXT -1 82 "it requ
ires nothing more than than gorey extra detail to get the full validit
y...)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "
Another question. And how many such " }{XPPEDIT 18 0 "alpha;" "6#%&alp
haG" }{TEXT -1 14 "s can one get?" }}{PARA 0 "" 0 "" {TEXT -1 185 "Imm
ediate answer, and observation. It's easy, and again - like I've alrea
dy pointed out in the Liouville section - there are an uncountable num
bers of such numbers: simply make up more " }{XPPEDIT 18 0 "alpha;" "6
#%&alphaG" }{TEXT -1 12 "s like this " }}{PARA 258 "" 0 "" {XPPEDIT
18 0 "alpha = Sum(a[m]/(10^(3^m)),m = 1 .. infinity);" "6#/%&alphaG-%$
SumG6$*&&%\"aG6#%\"mG\"\"\")\"#5)\"\"$F,!\"\"/F,;F-%)infinityG" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "where the sequence \{" }
{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 45 "\} is chosen as \+
in the Liouville section, and " }{TEXT 1231 3 "etc" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 27 " In fact, for any fixed " }{XPPEDIT
18 0 "kappa;" "6#%&kappaG" }{TEXT -1 46 " > 2, there are uncountably m
any real numbers " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 68 "
such that each of them has infinitely many rational approximations "
}{XPPEDIT 18 0 "p/q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 11 " satis
fying" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT
18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"
F," }{TEXT -1 3 " < " }{XPPEDIT 18 0 "1/(q^kappa);" "6#*&\"\"\"F$)%\"q
G%&kappaG!\"\"" }{TEXT -1 9 " ... (" }{XPPEDIT 18 0 "kappa;" "6#%&k
appaG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 68 "A Measure Theory motivated question. So, there are unc
ountably many " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 92 "s s
atisfying the previous inequality, but how much space do they take up \+
on the number line?" }}{PARA 0 "" 0 "" {TEXT -1 22 "Immediate answer. \+
Let " }{XPPEDIT 18 0 "S[kappa];" "6#&%\"SG6#%&kappaG" }{TEXT -1 54 " b
e the set of all real numbers for which inequality (" }{XPPEDIT 18 0 "
kappa;" "6#%&kappaG" }{TEXT -1 63 ") has an infinite number of rationa
l solutions, then [although " }{XPPEDIT 18 0 "S[kappa]" "6#&%\"SG6#%&k
appaG" }{TEXT -1 135 " appears to be large, and certainly is from a ca
rdinality point of view; in fact it has the same cardinality as the en
tire real line!] " }{XPPEDIT 18 0 "S[kappa];" "6#&%\"SG6#%&kappaG" }
{TEXT -1 50 " has Lebesgue measuere zero [and so appears to be " }
{TEXT 1233 11 "quite small" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Another question. Earlier it wa
s observed that the equation " }{XPPEDIT 18 0 "x^2-2*y^2 = 1;" "6#/,&*
$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(!\"\"F(" }{TEXT -1 63 " having an in
finite number of solutions in (positive) integers " }{TEXT 1234 1 "x"
}{TEXT -1 2 ", " }{TEXT 1235 2 "y " }{TEXT -1 5 "is a " }{TEXT 1236
15 "natural outcome" }{TEXT -1 19 " of observing that " }{XPPEDIT 18
0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 74 " is irrational; what ha
ppens if one replaces (e.g.) the irrational number " }{XPPEDIT 18 0 "s
qrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 28 " with the irrational number \+
" }{XPPEDIT 18 0 "2^(1/3);" "6#)\"\"#*&\"\"\"F&\"\"$!\"\"" }{TEXT -1
46 ", and what then is the effect on the equation " }{XPPEDIT 18 0 "x^
3-2*y^3 = 1;" "6#/,&*$%\"xG\"\"$\"\"\"*&\"\"#F(*$%\"yGF'F(!\"\"F(" }
{TEXT -1 41 " (since there are no (positive) integers " }{TEXT 1237 4
"x, y" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "x^3 = 2*y^3;" "6#/*$
%\"xG\"\"$*&\"\"#\"\"\"*$%\"yGF&F)" }{TEXT -1 13 ", and so the " }
{TEXT 1238 9 "next best" }{TEXT -1 18 " thing is to have " }{XPPEDIT
18 0 "x^3 = 2*y^3+1;" "6#/*$%\"xG\"\"$,&*&\"\"#\"\"\"*$%\"yGF&F*F*F*F*
" }{TEXT -1 8 ", i.e., " }{XPPEDIT 18 0 "x^3-2*y^3 = 1;" "6#/,&*$%\"xG
\"\"$\"\"\"*&\"\"#F(*$%\"yGF'F(!\"\"F(" }{TEXT -1 1 ")" }}{PARA 0 ""
0 "" {TEXT -1 11 "Every such " }{TEXT 1244 1 "x" }{TEXT -1 2 ", " }
{TEXT 1243 2 "y " }{TEXT -1 31 "would create a rational number " }
{XPPEDIT 18 0 "x/y;" "6#*&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 1 " " }
{TEXT 1246 8 "so close" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "2^(1/3);" "
6#)\"\"#*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 6 "as to " }{TEXT 1245 6 "almo
st" }{TEXT -1 32 " be a solution of the inequality" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha-p/q);" "6#-
%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"F," }{TEXT -1 3 " < " }
{XPPEDIT 18 0 "1/(q^3);" "6#*&\"\"\"F$*$%\"qG\"\"$!\"\"" }{TEXT -1 13
" ... (iii)" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 ""
{TEXT -1 32 "By Liouville's theorem all such " }{TEXT 1239 1 "x" }
{TEXT -1 2 ", " }{TEXT 1240 1 "y" }{TEXT -1 28 " must satisfy the ineq
uality" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT
18 0 "abs(2^(1/3)-x/y);" "6#-%$absG6#,&)\"\"#*&\"\"\"F*\"\"$!\"\"F**&%
\"xGF*%\"yGF,F," }{TEXT -1 1 " " }{TEXT 1241 1 ">" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c/(y^3);" "6#*&%\"cG\"\"\"*$%\"yG\"\"$!\"\"" }{TEXT -1
21 " , for some constant " }{TEXT 1242 1 "c" }{TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A big question th
en is: Does/doesn't the equation " }{XPPEDIT 18 0 "x^3-2*y^3 = 1" "6#/
,&*$%\"xG\"\"$\"\"\"*&\"\"#F(*$%\"yGF'F(!\"\"F(" }{TEXT -1 22 " (and o
thers like it: " }{XPPEDIT 18 0 "x^3-d*y^3 = 1;" "6#/,&*$%\"xG\"\"$\"
\"\"*&%\"dGF(*$%\"yGF'F(!\"\"F(" }{TEXT -1 10 ", general " }{TEXT
1250 8 "non-cube" }{TEXT -1 1 " " }{TEXT 1249 1 "d" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "x^4-2*y^4 = 1;" "6#/,&*$%\"xG\"\"%\"\"\"*&\"\"#F(*$%\"y
GF'F(!\"\"F(" }{TEXT -1 12 ", etc) have " }{TEXT 1251 3 "any" }{TEXT
-1 23 " solutions in integers " }{TEXT 1247 1 "x" }{TEXT -1 2 ", " }
{TEXT 1248 1 "y" }{TEXT -1 41 ", and if so does it have infinitely man
y?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "An
answer. We are now getting into very, very deep water, and an answer \+
will have to wait until we get to the Axel Thue section." }}}{SECT 1
{PARA 4 "" 0 "" {TEXT -1 18 "Cantor (1845-1918)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "One could spend one's wh
ole life thinking about, and talking about nothing else but Cantor. Ca
ntor, of whom Hilbert declared: " }{TEXT 349 73 "No one shall drive us
from the Paradise that Cantor has created for us..." }}{PARA 0 "" 0 "
" {TEXT -1 108 " Suffice it here to remark (as a gross understateme
nt!) that Cantor proved (1874) transcendental numbers " }{TEXT 347 7 "
existed" }{TEXT -1 50 "... and, in fact, that transcendental numbers a
re " }{TEXT 348 13 "more numerous" }{TEXT -1 174 " than algebraic numb
ers (strictly speaking that the (infinite) cardinality of the transcen
dental numbers is greater than the (infinite) cardinality of the algeb
raic numbers)." }}{PARA 0 "" 0 "" {TEXT -1 39 " Many readers will k
now of Cantor's " }{TEXT 351 14 "diagonal proof" }{TEXT -1 199 " that \+
the real numbers are uncountable, but not all may be familiar with his
vastly superior nested interval proof. For me (based on experience) t
he way to introduce Cantor's work is via the playful " }{TEXT 352 15 "
Hilbert's Hotel" }{TEXT -1 114 " (I used such an approach many years a
go, with non-mathematicians - in the mid 80's - when I used to offer y
early " }{TEXT 354 10 "extramural" }{TEXT -1 83 " courses at Universit
y College Dublin). I teach a second year undergraduate course " }
{TEXT 353 47 "The Real Number System and Cantorian Set Theory" }{TEXT
-1 188 " at my college, which the interested reader may read about at \+
my web site (there one will find: summary notes, exam paper, Maple tes
t (irreducible polynomials, continued fractions, etc)). " }}{PARA 0 "
" 0 "" {TEXT -1 55 " There is, of course, Cantor's revolutionary (1
873) " }{TEXT 417 9 "countable" }{TEXT -1 75 " enumeration of the rati
onals, with its slightly unpleasant casting out of " }{TEXT 415 10 "du
plicates" }{TEXT -1 106 "... In that connection some of my readers may
not be familiar with the quite wonderful Calkin-Wilf (2000) " }{TEXT
416 8 "explicit" }{TEXT -1 53 " enumeration (which, if you haven't see
n it, can you " }{TEXT 419 13 "figure it out" }{TEXT -1 76 "?) of the \+
positive rationals:\n " }
{XPPEDIT 18 0 "1/1,1/2,2/1,1/3,3/2,2/3,3/1,1/4,4/3,3/5,5/2,2/5,5/3,3/4
,4/1;" "61*&\"\"\"F$F$!\"\"*&F$F$\"\"#F%*&F'F$F$F%*&F$F$\"\"$F%*&F*F$F
'F%*&F'F$F*F%*&F*F$F$F%*&F$F$\"\"%F%*&F/F$F*F%*&F*F$\"\"&F%*&F2F$F'F%*
&F'F$F2F%*&F2F$F*F%*&F*F$F/F%*&F/F$F$F%" }{TEXT -1 52 ", ... \n\nwhich
is elaborated in Aigner and Ziegler's " }{TEXT 418 20 "Proofs from th
e BOOK" }{TEXT -1 80 ". A & Z write:\n\n \"Thus we have obtained a bea
utiful formula for the successor f(" }{TEXT 420 1 "x" }{TEXT -1 5 ") o
f " }{TEXT 421 1 "x" }{TEXT -1 181 " found recently by Moshe Newman ..
. \" \nAccording to Newman, the above Calkin-Wilf enumeration is the (
infinite sequence of) iterates (line 6 in following procedure) of the \+
function " }{XPPEDIT 18 0 "f(x) = 1/([x]+1-\{x\});" "6#/-%\"fG6#%\"xG*
&\"\"\"F),(7#F'F)F)F)<#F'!\"\"F-" }{TEXT -1 36 " (line 3), applied to \+
initial value " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 19
" (line 4), \nwhere [" }{TEXT 422 1 "x" }{TEXT -1 7 "] and \{" }{TEXT
423 1 "x" }{TEXT -1 10 "\} are the " }{TEXT 425 8 "integral" }{TEXT
-1 5 " and " }{TEXT 426 10 "fractional" }{TEXT -1 10 " parts of " }
{TEXT 424 1 "x" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 189 "CWNew := proc(bound)\nlocal f, r, k;\nf := x -> 1/(floor(x) + 1
- frac(x)):\nr[1] := 1/1:\n for k from 2 to bound do\n r[k] := \+
f(r[k-1])\nod: print(seq(r[k], k = 1..bound), `... etc`); end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "CWNew(50);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Hermite (1822-1901) (" }{TEXT
346 1 "e" }{TEXT -1 29 ") and Lindemann (1852-1939) (" }{XPPEDIT 18 0
"Pi;" "6#%#PiG" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 289 "While \+
it is true that Liouville gave, with proof (how else!), the first exam
ples of transcendental numbers, it would - I believe - be generally ac
cepted that the first example of a seriously beautiful transcendental \+
number is due to Charles Hermite, with his proof of the transcendence \+
of " }{TEXT 440 1 "e" }{TEXT -1 68 ", in 1873. (One can only imagine h
ow delighted he must have felt to " }{TEXT 444 4 "know" }{TEXT -1 83 "
, with complete certainty (in so far as such a thing exists), that Eul
er's number, " }{TEXT 441 1 "e" }{TEXT -1 5 ", is " }{TEXT 442 3 "not
" }{TEXT -1 15 " a solution of " }{TEXT 443 3 "any" }{TEXT -1 83 " pol
ynomial equation with integer coefficients, with non-zero leading coef
ficient.)" }}{PARA 0 "" 0 "" {TEXT -1 26 " His proof is actually "
}{TEXT 445 12 "quite simple" }{TEXT -1 124 " from a technical point of
view (it only required genius to frame it!), and can be easily follow
ed. Here I present only the " }{TEXT 446 5 "ideas" }{TEXT -1 177 " beh
ind his proof (the interested reader may follow the complete details b
y reading elsewhere, or in my Manchester 1972-1973 notes when I eventu
ally put them up at my web site)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "Basic idea of Hermite's proof of the tran
scendence of " }{TEXT 448 1 "e" }{TEXT -1 15 ". Suppose that " }{TEXT
447 1 "e" }{TEXT -1 19 " is algebraic. Then" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[m]*e^m+a[
m-1]*e^(m-1)+`...`+a[1]*e+a[0] = 0;" "6#/,,*&&%\"aG6#%\"mG\"\"\")%\"eG
F)F*F**&&F'6#,&F)F*F*!\"\"F*)F,,&F)F*F*F1F*F*%$...GF**&&F'6#F*F*F,F*F*
&F'6#\"\"!F*F:" }{TEXT -1 11 " ... (i)" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "for some integers " }{XPPEDIT 18
0 "a[m],a[m-1],`...`+a[1],a[0];" "6&&%\"aG6#%\"mG&F$6#,&F&\"\"\"F*!\"
\",&%$...GF*&F$6#F*F*&F$6#\"\"!" }{TEXT -1 7 ", with " }{XPPEDIT 18 0
"a[m] <> 0;" "6#0&%\"aG6#%\"mG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 49 " Hermite's idea was to replace every power of " }
{TEXT 639 1 "e" }{TEXT 640 1 " " }{TEXT -1 12 "in (i) with " }{TEXT
449 12 "simultaneous" }{TEXT -1 48 " rational approximations (meaning \+
they have the " }{TEXT 450 4 "same" }{TEXT -1 19 " denominator) of a \+
" }{TEXT 638 17 "very special kind" }{TEXT -1 76 "; he showed how to c
onstruct single-variable rational number approximations " }{XPPEDIT
18 0 "F(m)/F(0),F(m-1)/F(0),`...`,F(1)/F(0);" "6&*&-%\"FG6#%\"mG\"\"\"
-F%6#\"\"!!\"\"*&-F%6#,&F'F(F(F,F(-F%6#F+F,%$...G*&-F%6#F(F(-F%6#F+F,
" }{TEXT -1 8 " (with " }{TEXT 641 18 "increasingly large" }{TEXT -1
20 " common denominator " }{XPPEDIT 18 0 "F(0);" "6#-%\"FG6#\"\"!" }
{TEXT -1 17 ") to the numbers " }{XPPEDIT 18 0 "e^m,e^(m-1),`...`,e;"
"6&)%\"eG%\"mG)F$,&F%\"\"\"F(!\"\"%$...GF$" }{TEXT -1 22 ", such that \+
(i) became" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "a[m]*(F(m)/F(0)+epsilon[m])+a[m-1]*(F(m-1)/F(0)+epsilon
[m-1])+`...`+a[1]*(F(1)/F(0)+epsilon[1])+a[0] = 0;" "6#/,,*&&%\"aG6#%
\"mG\"\"\",&*&-%\"FG6#F)F*-F.6#\"\"!!\"\"F*&%(epsilonG6#F)F*F*F**&&F'6
#,&F)F*F*F3F*,&*&-F.6#,&F)F*F*F3F*-F.6#F2F3F*&F56#,&F)F*F*F3F*F*F*%$..
.GF**&&F'6#F*F*,&*&-F.6#F*F*-F.6#F2F3F*&F56#F*F*F*F*&F'6#F2F*F2" }
{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 6 "namely" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "a[m]*F(m)+a[m-1
]*F(m-1)+`...`+a[1]*F(1)+a[0]*F(0);" "6#,,*&&%\"aG6#%\"mG\"\"\"-%\"FG6
#F(F)F)*&&F&6#,&F(F)F)!\"\"F)-F+6#,&F(F)F)F1F)F)%$...GF)*&&F&6#F)F)-F+
6#F)F)F)*&&F&6#\"\"!F)-F+6#F>F)F)" }{TEXT -1 8 "\n + " }{XPPEDIT
18 0 "F(0)*a[m]*epsilon[m]+F(0)*a[m-1]*epsilon[m-1]+`...`+F(0)*a[1]*ep
silon[1] = 0;" "6#/,**(-%\"FG6#\"\"!\"\"\"&%\"aG6#%\"mGF*&%(epsilonG6#
F.F*F**(-F'6#F)F*&F,6#,&F.F*F*!\"\"F*&F06#,&F.F*F*F8F*F*%$...GF**(-F'6
#F)F*&F,6#F*F*&F06#F*F*F*F)" }{TEXT -1 10 " ... (ii)" }}{PARA 0 "" 0
"" {TEXT -1 79 " Hermite was able to arrange the rational approxima
tions in such a way that " }{TEXT 643 8 "not only" }{TEXT -1 22 " are \+
all the epsilons " }}{PARA 0 "" 0 "" {TEXT 644 5 "small" }{TEXT -1
144 ", and become increasingly smaller as F(0) is made increasingly la
rge (in itself that is an entirely trivial matter in a general setting
: choose " }{TEXT 452 3 "any" }{TEXT -1 1 " " }{TEXT 451 1 "m" }{TEXT
-1 14 " real numbers " }{XPPEDIT 18 0 "alpha[1],alpha[2],`...`,alpha[m
];" "6&&%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"mG" }{TEXT -1 6 ", and
" }{TEXT 454 3 "any" }{TEXT -1 13 " denominator " }{TEXT 453 1 "q" }
{TEXT -1 15 " (and think of " }{TEXT 642 1 "q" }{TEXT -1 54 " as being
made larger and larger), then each of those " }{XPPEDIT 18 0 "alpha;
" "6#%&alphaG" }{TEXT -1 48 "'s is either a rational number with denom
inator " }{TEXT 458 1 "q" }{TEXT -1 68 ", or lies between two consecut
ive rational numbers with denominator " }{TEXT 459 1 "q" }{TEXT -1 34
". Thus there are rational numbers " }{XPPEDIT 18 0 "f(1)/q,f(2)/q,`..
.`,f(m)/q;" "6&*&-%\"fG6#\"\"\"F'%\"qG!\"\"*&-F%6#\"\"#F'F(F)%$...G*&-
F%6#%\"mGF'F(F)" }{TEXT -1 50 " that are simultaneous rational approxi
mations to " }{XPPEDIT 18 0 "alpha[1], alpha[2], `...`, alpha[m]" "6&&
%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"mG" }{TEXT -1 44 ", with, in e
ach case, an error term at most " }{XPPEDIT 18 0 "1/q;" "6#*&\"\"\"F$%
\"qG!\"\"" }{TEXT -1 8 " ), but " }{TEXT 456 8 "so small" }{TEXT -1 6
" that " }{TEXT 457 9 "every one" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "F
(0)*epsilon[m],F(0)*epsilon[m-1],`...`,F(0)*epsilon[1];" "6&*&-%\"FG6#
\"\"!\"\"\"&%(epsilonG6#%\"mGF(*&-F%6#F'F(&F*6#,&F,F(F(!\"\"F(%$...G*&
-F%6#F'F(&F*6#F(F(" }{TEXT -1 4 " is " }{TEXT 455 5 "small" }{TEXT -1
70 ", and become increasingly smaller as F(0) is made increasingly lar
ge. " }}{PARA 0 "" 0 "" {TEXT -1 19 " You should now " }{TEXT 645
3 "see" }{TEXT -1 10 " how (ii) " }{TEXT 646 5 "reads" }{TEXT -1 127 "
: it looks like (an integer, which varies) + (something small, that's \+
getting smaller) = 0. That, however, would be impossible " }{TEXT 647
7 "if only" }{TEXT -1 180 " one could arrange matters so that the 'int
eger' is non-zero. In short, that's what Hermite did, but it in a quit
e complicated way... It was greatly simplified (by Klein?) with an " }
{TEXT 648 6 "ad hoc" }{TEXT -1 33 " piece of trickery: arrange for ("
}{XPPEDIT 18 0 "a[m]*F(m)+a[m-1]*F(m-1)+`...`+a[1]*F(1)+a[0]*F(0)" "6#
,,*&&%\"aG6#%\"mG\"\"\"-%\"FG6#F(F)F)*&&F&6#,&F(F)F)!\"\"F)-F+6#,&F(F)
F)F1F)F)%$...GF)*&&F&6#F)F)-F+6#F)F)F)*&&F&6#\"\"!F)-F+6#F>F)F)" }
{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 42 "to be non-zero by choos
ing a prime number " }{TEXT 649 1 "p" }{TEXT -1 22 " that does not div
ide " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 43 ", and arr
ange the approximations such that " }{TEXT 650 1 "p" }{TEXT -1 6 " doe
s " }{TEXT 651 3 "not" }{TEXT -1 34 " divide the denominator F(0), but
" }{TEXT 652 4 "does" }{TEXT -1 36 " divide every one of the numerato
rs " }{XPPEDIT 18 0 "F(m),F(m-1),`...`,F(1);" "6&-%\"FG6#%\"mG-F$6#,&F
&\"\"\"F*!\"\"%$...G-F$6#F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT
-1 358 " A novice reader should seek out the actual detailed proof.
Some versions are not for the faint hearted; the one I first read in \+
Hardy & Wright is quite frightful, and certainly does not aim to enlig
hten. My own recollection of understanding the proof for the first tim
e was reading my school-bought copy of Felix Klein's (Dover edition, w
hich I've lost) " }{TEXT 653 10 "Arithmetic" }{TEXT -1 226 "; that doe
sn't seem to be available anymore. I had hoped to type up my Mancheste
r hand-written notes in time for this talk, but that will have to wait
. Interested readers ought to find they follow the proof I gave there.
" }}{PARA 258 "" 0 "" {TEXT -1 9 "_________" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Pi;" "6#
%#PiG" }{TEXT -1 152 " is so well known to everyone that it would not \+
be sensible for me to write about it here. The following are classic, \+
and well known: area of a circle (" }{XPPEDIT 18 0 "A = Pi*r^2;" "6#/%
\"AG*&%#PiG\"\"\"*$%\"rG\"\"#F'" }{TEXT -1 30 "), circumference of a c
ircle (" }{XPPEDIT 18 0 "C = 2*pi*r;" "6#/%\"CG*(\"\"#\"\"\"%#piGF'%\"
rGF'" }{TEXT -1 22 "), volume of a sphere " }}{PARA 0 "" 0 "" {TEXT
-1 1 "(" }{XPPEDIT 18 0 "V = 4*Pi*r^3/3;" "6#/%\"VG**\"\"%\"\"\"%#PiGF
'%\"rG\"\"$F*!\"\"" }{TEXT -1 30 " ), surface area of a sphere (" }
{XPPEDIT 18 0 "S = 4*Pi*r^2;" "6#/%\"SG*(\"\"%\"\"\"%#PiGF'%\"rG\"\"#
" }{TEXT -1 7 " ), etc" }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "zeta(2);" "6
#-%%zetaG6#\"\"#" }{TEXT -1 4 " := " }{XPPEDIT 18 0 "Sum(1/(n^2),n = 1
.. infinity) = 1/(1^2)+1/(2^2)+1/(3^3)+1/(4^2);" "6#/-%$SumG6$*&\"\"
\"F(*$%\"nG\"\"#!\"\"/F*;F(%)infinityG,**&F(F(*$F(F+F,F(*&F(F(*$F+F+F,
F(*&F(F(*$\"\"$F7F,F(*&F(F(*$\"\"%F+F,F(" }{TEXT -1 16 " + ... , equal
s " }{XPPEDIT 18 0 "Pi^2/6" "6#*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 31 ",
as was first proved by Euler." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 37 "restart;\nsum(1/n^2, n = 1..infinity);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 113 "If you aren't familiar with what a CAS - like Maple
- can do, I hope you are impressed with that last evaluation!" }}
{PARA 15 "" 0 "" {TEXT -1 103 "the probability (properly defined) that
two random integers have greatest common divisor equal to 1 is " }
{XPPEDIT 18 0 "6/(Pi^2);" "6#*&\"\"'\"\"\"*$%#PiG\"\"#!\"\"" }{TEXT
-1 39 "\n(related to the previous infinite sum)" }}{PARA 15 "" 0 ""
{TEXT -1 32 "the zeroes of the function of a " }{TEXT 898 7 "complex"
}{TEXT -1 14 " variable sin(" }{TEXT 897 1 "z" }{TEXT -1 83 "), with i
nfinite series expansion\n\n \+
" }{XPPEDIT 18 0 "Sum((-1)^(n+1)*z^(2*n-1)/(2*n-1)!,n = 1 .. inf
inity) = z-z^3/3!+z^5/5!-z^7/7!;" "6#/-%$SumG6$*(),$\"\"\"!\"\",&%\"nG
F*F*F*F*)%\"zG,&*&\"\"#F*F-F*F*F*F+F*-%*factorialG6#,&*&F2F*F-F*F*F*F+
F+/F-;F*%)infinityG,*F/F**&F/\"\"$-F46#F=F+F+*&F/\"\"&-F46#FAF+F**&F/
\"\"(-F46#FEF+F+" }{TEXT -1 16 " + ...\n\nare ... " }{XPPEDIT 18 0 "-4
*Pi,-2*Pi,0,2*Pi,4*Pi,6*Pi;" "6(,$*&\"\"%\"\"\"%#PiGF&!\"\",$*&\"\"#F&
F'F&F(\"\"!*&F+F&F'F&*&F%F&F'F&*&\"\"'F&F'F&" }{TEXT -1 5 ", ..." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The grea
t Siegel began his classic 1949 Princeton University Press on Transcen
dental Numbers by writing " }{TEXT 901 79 "The most widely known resul
t on transcendental numbers is the transcendency of " }{XPPEDIT 18 0 "
Pi;" "6#%#PiG" }{TEXT 902 29 " proved by Lindemann in 1882." }{TEXT
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 334 "If one is acquainted with the
classic Greek ruler-and-compass construction problems (sadly not a to
pic that our modern school pupils are exposed to...) of the duplicatio
n of the cube, the trisection of a general angle, and the squaring of \+
the circle, then one will know that a solution of the latter reduces t
o knowing whether or not " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1
70 " is not just algebraic, but is algebraic of a very particular kind
... " }}{PARA 0 "" 0 "" {TEXT -1 73 " I think I should be frank and
admit that there is no easy proof that " }{XPPEDIT 18 0 "Pi;" "6#%#Pi
G" }{TEXT -1 357 " is transcendental. I did include a fairly clear pro
of of it in my Manchester 1972-73 course, and hope in time to put it u
p in the transcendental numbers corner of my web site. I will merely r
ecord that the honour of first proving its transcendence goes to Linde
mann (1882), and recommend that interested readers consider obtaining \+
a copy of the delightful " }{TEXT 899 2 "Pi" }{TEXT 900 16 ": A Source
Book " }{TEXT -1 42 "(see References). " }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 "
Thue, Mordell, Siegel, Mahler, Schneider, Roth, Schmidt" }}{SECT 1
{PARA 4 "" 0 "" {TEXT -1 21 "Axel Thue (1863-1922)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 437 21 "Thue's Theorem (1909)" }
{TEXT -1 6 ". Let " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 40
" be any real algebraic number of degree " }{TEXT 366 1 "n" }{TEXT -1
2 " (" }{TEXT 370 8 "at least" }{TEXT -1 13 " 3), and let " }{XPPEDIT
18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 27 " be any positive constant
(" }{TEXT 371 13 "however small" }{TEXT -1 37 "), then there is a pos
itive constant " }{XPPEDIT 18 0 "c = c(alpha,epsilon);" "6#/%\"cG-F$6$
%&alphaG%(epsilonG" }{TEXT -1 20 " (i.e. the value of " }{TEXT 367 1 "
c" }{TEXT -1 9 " depends " }}{PARA 0 "" 0 "" {TEXT 368 4 "only" }
{TEXT -1 4 " on " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 11 ") such tha
t" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%
&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"F," }{TEXT -1 1 " " }{TEXT 369 1 ">" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "c(alpha)/(q^(n/2+1+epsilon));" "6#*&-%
\"cG6#%&alphaG\"\"\")%\"qG,(*&%\"nGF(\"\"#!\"\"F(F(F(%(epsilonGF(F/" }
{TEXT -1 11 " ... (i')" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 91 "One can only write of the fundamental importance o
f that result by resorting to hyperbole. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "I highly recommend Wolfgang Schmid
t's AMS Bulletin (1978) review of " }{TEXT 372 41 "Selected mathematic
al papers of Axel Thue" }{TEXT -1 21 ", from which I quote:" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 373
214 "\" His greatest work, on approximation to algebraic numbers, appe
ared in 1908/1909, when he was well in his forties, and when he had be
en away from the centres of mathematics for over a decade. ... Landau \+
called it " }{TEXT -1 55 "[the above theorem, and its application to t
he related " }{TEXT 374 13 "Thue equation" }{TEXT -1 2 "] " }{TEXT
375 206 "the most important discovery in elementary number theory whic
h he had witnessed in his lifetime. He also said ten years after its p
ublication that already ten competent mathematicians had read Thue's p
aper." }{TEXT 436 1 "\"" }{TEXT -1 209 " [JC comment. Far be it from m
e to correct Schmidt's English (his first language is German) but I th
ink any reader will believe that what Landau said, ten years after the
publication of Thue's paper, was that " }{TEXT 376 4 "only" }{TEXT
-1 21 " ten competent... ]\n\n" }{TEXT 377 81 "\" Siegel, trying to un
derstand Thue's paper, rewrote it , and in the process ...\"" }{TEXT
-1 74 " [introduced refinements which improved the above (i'), see nex
t section]\n" }{TEXT 378 1 "\n" }{TEXT 439 162 "\" ... Thue's papers a
re sometimes difficult to read, not in the least because often the rea
der does not know until the end where the investigation is leading to.
\"" }{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 10 "__________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "A fundame
ntal (and easy) consequence of Thue's theorem. Let" }}{PARA 258 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y) = a[0]*x^n+a[1]*x^(n-1)*y+`...
`+a[n-1]*x*y^(n-1)+a[n]*y^n;" "6#/-%\"fG6$%\"xG%\"yG,,*&&%\"aG6#\"\"!
\"\"\")F'%\"nGF/F/*(&F,6#F/F/)F',&F1F/F/!\"\"F/F(F/F/%$...GF/*(&F,6#,&
F1F/F/F7F/F'F/)F(,&F1F/F/F7F/F/*&&F,6#F1F/)F(F1F/F/" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 53 "be a homogeneous form of degree at least
3 such that " }{XPPEDIT 18 0 "f(x,1);" "6#-%\"fG6$%\"xG\"\"\"" }
{TEXT -1 53 " has no repeated zero, then the Diophantine equation " }
{XPPEDIT 18 0 "f(x,y) = m;" "6#/-%\"fG6$%\"xG%\"yG%\"mG" }{TEXT -1 47
" has only finitely many solutions for integral " }{XPPEDIT 18 0 "m <>
0;" "6#0%\"mG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 " \+
In particular the cubic and higher degree versions of the classic "
}{TEXT 385 11 "Fermat-Pell" }{TEXT -1 10 " equation " }{XPPEDIT 18 0 "
x^2-d*y^2 = 1;" "6#/,&*$%\"xG\"\"#\"\"\"*&%\"dGF(*$%\"yGF'F(!\"\"F(" }
{TEXT -1 13 " (which have " }{TEXT 387 10 "infinitely" }{TEXT -1 61 " \+
many integral solutions for every non-square natural number " }{TEXT
386 1 "d" }{TEXT -1 14 ") have only a " }{TEXT 388 6 "finite" }{TEXT
-1 21 " number of solutions:" }}{PARA 15 "" 0 "" {TEXT -1 18 "For ever
y integer " }{TEXT 391 1 "d" }{TEXT -1 16 ", the equation " }
{XPPEDIT 18 0 "x^3-d*y^3 = 1" "6#/,&*$%\"xG\"\"$\"\"\"*&%\"dGF(*$%\"yG
F'F(!\"\"F(" }{TEXT -1 12 " has only a " }{TEXT 392 6 "finite" }{TEXT
-1 33 " number of solutions in integers " }{TEXT 389 1 "x" }{TEXT -1
5 " and " }{TEXT 390 1 "y" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT
-1 18 "For every integer " }{TEXT 394 1 "d" }{TEXT -1 16 ", the equati
on " }{XPPEDIT 18 0 "x^4-d*y^4 = 1;" "6#/,&*$%\"xG\"\"%\"\"\"*&%\"dGF
(*$%\"yGF'F(!\"\"F(" }{TEXT -1 12 " has only a " }{TEXT 395 6 "finite
" }{TEXT -1 33 " number of solutions in integers " }{TEXT 393 1 "x" }
{TEXT -1 5 " and " }{TEXT 396 1 "y" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT
-1 30 "Louis Joel Mordell (1888-1972)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 785 22 "This small section \+
is " }{TEXT 788 11 "essentially" }{TEXT 789 26 " about the deep questi
on:\n" }{TEXT 786 49 "how close can squares and cubes get to each othe
r" }{TEXT 787 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 185 "You may view a photograph of Mordell in the Oxford 1969 \+
corner of my web site, and a short account of a conversation I had wit
h him at the time the photograph was taken. Here I mention " }{TEXT
769 8 "only one" }{TEXT -1 164 " famous result of Mordell - which is i
ntimately related to Thue's theorem - and here mention that both Thue \+
and Mordell relate to later profound work of Alan Baker." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "Some (fast-track!
) background. Everyone knows the squares: 0, 1, 4, 9, 16, ... , and it
is trivial to explain why the successive differences between them, na
mely 1, 3, 5, 7, ... , follow a " }{TEXT 756 11 "predictable" }{TEXT
-1 9 " pattern:" }}{PARA 0 "" 0 "" {TEXT -1 17 "it's simply that " }
{XPPEDIT 18 0 "(n+1)^2-n^2 = n^2+2*n+1-n^2;" "6#/,&*$,&%\"nG\"\"\"F(F(
\"\"#F(*$F'F)!\"\",**$F'F)F(*&F)F(F'F(F(F(F(*$F'F)F+" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "2*n+1;" "6#,&*&\"\"#\"\"\"%\"nGF&F&F&F&" }{TEXT -1
33 ", and the terms of the sequence \{" }{XPPEDIT 18 0 "2*n+1;" "6#,&*
&\"\"#\"\"\"%\"nGF&F&F&F&" }{TEXT -1 255 "\} are merely 1, 3, 5, 7, ..
. . The same sort of thing happens with cubes, though now the first se
quence of differences behaves quadratically, while the second sequence
of differences behaves linearly:\n\nThe cubes are: 0, 1, 8, 27
, 64, 125, 216, ... " }}{PARA 0 "" 0 "" {TEXT -1 47 "The differences: \+
1, 7, 19, 37, 61, 91, ... " }}{PARA 0 "" 0 "" {TEXT -1 90 "Their d
ifferences: 6, 12, 18, 24, 30, ... [and it's an easy exercise to ex
plain why...]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 22 "But now for something " }{TEXT 760 15 "quite different" }
{TEXT -1 168 ". The mixed sequence of all squares and cubes (the ones \+
in the larger font size are those that are both a square and a cube, a
nd so are a sixth power) starts like this:" }}{PARA 258 "" 0 "" {TEXT
757 1 "0" }{TEXT -1 2 ", " }{TEXT 758 1 "1" }{TEXT -1 31 ", 4, 8, 9, 1
6, 25, 27, 36, 49, " }{TEXT 759 2 "64" }{TEXT -1 24 ", 81, 100, 121, 1
25, ..." }}{PARA 0 "" 0 "" {TEXT -1 299 " Now look at the successiv
e differences: 1, 3, 4, 1, 7, 9, 2, 9, 13, ... . As you see, the diffe
rences are most certainly not monotonic - as was the case with the squ
ares and cubes separately - and jump about all over the place ('live' \+
I can vary the 'bound' - I'll probably make it be 100000 (a " }{TEXT
784 4 "lakh" }{TEXT -1 127 " in India) - just to allow the third '17' \+
difference to be seen; but in the fixed html version I will probably s
et it at 1000):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "bound :=
1000: \nSs := \{\}: for k from 0 while k^2 <= bound do\nSs := \{op(Ss
), k^2\} od: Ss:\nSc := \{\}: for k from 0 while k^3 <= bound do\nSc :
= \{op(Sc), k^3\} od: Sc:\nmixed := Ss union Sc; N := nops(mixed):\ndi
fferences := [seq(mixed[k+1] - mixed[k], k=1..N-1)];\ndiffer_occur := \+
sort(differences);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Some (fas
t-track) observations and questions (see the Alan Baker section later)
." }}{PARA 0 "" 0 "" {TEXT -1 263 "First a trivial ovservation: it is \+
completely elementary to determine all possible integral square that d
iffer by a given amount. For example, let's find all squares differing
by (say) 15. I omit all obvious comment concerning 15 being minus 5 t
imes minus 3 etc. " }{XPPEDIT 18 0 "y^2-x^2 = 15;" "6#/,&*$%\"yG\"\"#
\"\"\"*$%\"xGF'!\"\"\"#:" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(y-x)*(y+x)
= 15;" "6#/*&,&%\"yG\"\"\"%\"xG!\"\"F',&F&F'F(F'F'\"#:" }{TEXT -1 22
", gives possibilities:" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 790
8 "y + x = " }{TEXT -1 2 "15" }{TEXT 791 1 " " }{TEXT -1 4 "and " }
{XPPEDIT 18 0 "y-x = 1;" "6#/,&%\"yG\"\"\"%\"xG!\"\"F&" }{TEXT -1 8 " \+
, thus " }{TEXT 792 1 "y" }{TEXT -1 9 " = 8 and " }{TEXT 793 1 "x" }
{TEXT -1 4 " = 7" }}{PARA 15 "" 0 "" {TEXT 794 8 "y + x = " }{TEXT -1
1 "5" }{TEXT 797 1 " " }{TEXT -1 4 "and " }{XPPEDIT 18 0 "y-x = 3;" "6
#/,&%\"yG\"\"\"%\"xG!\"\"\"\"$" }{TEXT -1 8 " , thus " }{TEXT 795 1 "y
" }{TEXT -1 9 " = 4 and " }{TEXT 796 1 "x" }{TEXT -1 4 " = 1" }}{PARA
0 "" 0 "" {TEXT -1 296 "And thus 49 and 64 (which are consecutive), an
d 1 and 16 are the only squares that differ by 15. A similar check wil
l reveal the unsurprising fact that there are no squares that differ b
y (e.g.) 8. A similar analysis may be made for cubes differing by a gi
ven amount; one uses the factorisation " }{XPPEDIT 18 0 "y^3-x^3 = (y
-x)*(y^2+y*x+x^2);" "6#/,&*$%\"yG\"\"$\"\"\"*$%\"xGF'!\"\"*&,&F&F(F*F+
F(,(*$F&\"\"#F(*&F&F(F*F(F(*$F*F0F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0
"" {TEXT -1 126 " But it's an entirely different matter for the mix
ed squares and cubes (and do bear in mind that there are all those oth
er " }{TEXT 798 9 "difficult" }{TEXT -1 68 " variations: squares and f
irst powers, cubes and fifth powers, ... )" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Observation I. 8 and 9 are a cu
be and square differing by 1. Are there any others? In fact, what " }
{TEXT 761 3 "are" }{TEXT -1 31 " the integers that satisfy the " }
{TEXT 762 12 "diophantine " }{TEXT -1 9 "equation " }{XPPEDIT 18 0 "y^
2 = x^3+1;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"F+F+" }{TEXT -1 28 "? \+
And the same question for " }{XPPEDIT 18 0 "y^2 = x^3-1;" "6#/*$%\"yG
\"\"#,&*$%\"xG\"\"$\"\"\"F+!\"\"" }{TEXT -1 242 ". It happens - but it
is not easy to prove that, apart from the trivial 0 and 1, 8 and 9 ar
e the only square and cube that differ by 1. It's a recently settled -
and much hailed result - that 8 and 9 are the only proper powers diff
ering by 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 181 "Observation II. 25 and 27 are a square and a cube differing by
2. Are there any others? Fermat reckoned - and asked around for a pro
of - that they were the only ones. In fact, what " }{TEXT 763 3 "are"
}{TEXT -1 31 " the integers that satisfy the " }{TEXT 764 12 "Diophant
ine " }{TEXT -1 9 "equation " }{XPPEDIT 18 0 "y^2 = x^3+2;" "6#/*$%\"y
G\"\"#,&*$%\"xG\"\"$\"\"\"F&F+" }{TEXT -1 28 ", and the same question \+
for " }{XPPEDIT 18 0 "y^2 = x^3-2;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"
\"F&!\"\"" }{TEXT -1 21 ". Answers are known.\n" }}{PARA 0 "" 0 ""
{TEXT -1 195 "Observation III. 16 and 25 are squares differing by 9, w
hile 27 and 36 are a cube and square differing also by 9, and (further
along) 216 and 225 are another cube and square differing by 9. What \+
" }{TEXT 765 3 "are" }{TEXT -1 31 " the integers that satisfy the " }
{TEXT 766 12 "diophantine " }{TEXT -1 9 "equation " }{XPPEDIT 18 0 "y^
2 = x^3+9;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"\"\"*F+" }{TEXT -1 28
", and the same question for " }{XPPEDIT 18 0 "y^2 = x^3-9;" "6#/*$%\"
yG\"\"#,&*$%\"xG\"\"$\"\"\"\"\"*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "It is entirely trivial
to find all integer solutions of (say) " }{XPPEDIT 18 0 "y^2 = x^2+9;
" "6#/*$%\"yG\"\"#,&*$%\"xGF&\"\"\"\"\"*F*" }{TEXT -1 33 " (that is no
t a typing error in '" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT
-1 35 "') or all the integer solutions of " }{XPPEDIT 18 0 "y^3 = x^3+
9;" "6#/*$%\"yG\"\"$,&*$%\"xGF&\"\"\"\"\"*F*" }{TEXT -1 8 " (those " }
{TEXT 767 3 "are" }{TEXT -1 75 " cubes), but quite a different matter \+
to determine all solutions of either " }{XPPEDIT 18 0 "y^2 = x^3+9;" "
6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"\"\"*F+" }{TEXT -1 5 " or " }
{XPPEDIT 18 0 "y^2 = x^3-9;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"\"\"*
!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 66 "Mordell's Theorem (1914 and 1922). For any fixed non-z
ero integer " }{TEXT 753 1 "k" }{TEXT -1 41 " [positive or negative] t
here are only a " }{TEXT 768 6 "finite" }{TEXT -1 20 " number of integ
ers " }{TEXT 754 1 "x" }{TEXT -1 5 " and " }{TEXT 755 1 "y" }{TEXT -1
11 " such that " }{XPPEDIT 18 0 "y^2 = x^3+k;" "6#/*$%\"yG\"\"#,&*$%\"
xG\"\"$\"\"\"%\"kGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 295 "Mordell used Thue's 1909 theorem in a \+
fundamental way in his proof; Mordell was unaware of Thue's result in \+
1914, and only came to hear of it much later (remember Landau's remark
). A fundamental weakness in Mordell's result - which itself is relate
d to the same in Thue's proof - is that it is '" }{TEXT 770 11 "ineffe
ctive" }{TEXT -1 142 "', meaning that while it tells us that there are
only a finite number of solutions, it doesn't allow us to know when w
e've found all of them, " }{TEXT 771 2 "if" }{TEXT -1 26 " there are a
ny, and there " }{TEXT 772 3 "may" }{TEXT -1 75 " be none (a nice exam
ple where there are none is provided by the equation " }{XPPEDIT 18
0 "y^2 = x^3+7;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"\"\"(F+" }{TEXT
-1 102 ", which can be proved in a reasonably elementary way, as I som
etimes show to my second year students)." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "Finally, and to give just a flavo
ur of a classic hard-fought result, I mention a theorem of Nagell's (1
930): the Diophantine equation " }{XPPEDIT 18 0 "y^2 = x^3+17;" "6#/*$
%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"\"#" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c(alpha)/(q^(2*sqrt(n)+epsilon));" "6#*&-%\"cG6#%&alpha
G\"\"\")%\"qG,&*&\"\"#F(-%%sqrtG6#%\"nGF(F(%(epsilonGF(!\"\"" }{TEXT
-1 12 " ... (i'')" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 11 "and Siegel " }{TEXT 405 11 "conjectured" }{TEXT -1 36 "
that the above inequality could be " }{TEXT 406 8 "improved" }{TEXT
-1 3 " to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%
\"qG!\"\"F," }{TEXT -1 1 " " }{TEXT 404 1 ">" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c(alpha)/(q^(2+epsilon));" "6#*&-%\"cG6#%&alphaG\"\"\")
%\"qG,&\"\"#F(%(epsilonGF(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "which, if true, would, o
f course, be best possible (in a sense), since by a classic (and easy \+
to prove theorem of Dirichlet) one has: for every real irrational numb
er there are infinitely many distinct rational numbers " }{XPPEDIT 18
0 "p/q;" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 2 " (" }{TEXT 407 1 "p
" }{TEXT -1 5 " and " }{TEXT 408 1 "q" }{TEXT -1 16 " integers, with \+
" }{XPPEDIT 18 0 "0 < q;" "6#2\"\"!%\"qG" }{TEXT -1 11 ") such that" }
}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha-p/q) < 1/(q^2);" "6#2-%$ab
sG6#,&%&alphaG\"\"\"*&%\"pGF)%\"qG!\"\"F-*&F)F)*$F,\"\"#F-" }{TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Si
egel was the first to prove transcendence results involving elliptic f
unctions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
712 23 "Siegel's (1932) Theorem" }{TEXT -1 8 ". Let P(" }{TEXT 713 1 "
z" }{TEXT -1 60 ") be the Weirstrass elliptic function with algebraic \+
number " }{TEXT 714 10 "invariants" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[
2];" "6#&%\"gG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[3];" "6#&
%\"gG6#\"\"$" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "diff(P(
z),z)^2 = 4*P(z)^3-g[2]*P(z)-g[3];" "6#/*$-%%diffG6$-%\"PG6#%\"zGF+\"
\"#,(*&\"\"%\"\"\"*$-F)6#F+\"\"$F0F0*&&%\"gG6#F,F0-F)6#F+F0!\"\"&F76#F
4F;" }{TEXT -1 36 " (the standard connection between P(" }{TEXT 715 1
"z" }{TEXT -1 25 ") and its derivative P '(" }{TEXT 716 1 "z" }{TEXT
-1 29 ")), then at least one of any " }{TEXT 923 11 "fundamental" }
{TEXT -1 22 " pair of periods of P(" }{TEXT 717 1 "z" }{TEXT -1 21 ") \+
is transcendental. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 36 "At the end of chapter 3 of his book " }{TEXT 1085 22 "T
ranscendental Numbers" }{TEXT -1 182 ", the one dealing with Hilbert's
seventh problem, Siegel wrote (to illustrate the state of ignorance, \+
I have slightly changed Siegel's notation and wording, and made some e
mphases): " }{TEXT 1086 35 "The result on the transcendency of " }
{XPPEDIT 18 0 "alpha^beta;" "6#)%&alphaG%%betaG" }{TEXT 1087 33 " can \+
also be stated this way: If " }{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&
%&alphaG6#\"\"\"&F$6#\"\"#" }{TEXT -1 1 " " }{TEXT 1088 23 "are algebr
aic numbers, " }{XPPEDIT 18 0 "alpha[1]*alpha[2] <> 0;" "6#0*&&%&alpha
G6#\"\"\"F(&F&6#\"\"#F(\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "log(alp
ha[1]) <> 0;" "6#0-%$logG6#&%&alphaG6#\"\"\"\"\"!" }{TEXT -1 2 ", " }
{TEXT 1089 15 "then the ratio " }{XPPEDIT 18 0 "log(alpha[2])/log(alph
a[1]);" "6#*&-%$logG6#&%&alphaG6#\"\"#\"\"\"-F%6#&F(6#F+!\"\"" }{TEXT
-1 1 " " }{TEXT 1090 177 "is either rational or transcendental. In oth
er words, the logarithm of any algebraic number relative to any algebr
aic base is either rational or transcendental. However...it is " }
{TEXT 1096 14 "not even known" }{TEXT 1097 15 " whether there " }
{TEXT 1091 6 "cannot" }{TEXT 1092 40 " exist an inhomogeneous linear r
elation " }{XPPEDIT 18 0 "beta[1]*log(alpha[1])+beta[2]*log(alpha[2]) \+
= 1;" "6#/,&*&&%%betaG6#\"\"\"F)-%$logG6#&%&alphaG6#F)F)F)*&&F'6#\"\"#
F)-F+6#&F.6#F3F)F)F)" }{TEXT -1 1 " " }{TEXT 1093 4 "with" }{TEXT -1
1 " " }{TEXT 1094 20 "quadratic irrational" }{TEXT -1 1 " " }{TEXT
1095 0 "" }{XPPEDIT 18 0 "beta[1],beta[2];" "6$&%%betaG6#\"\"\"&F$6#\"
\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 142 " And that ques
tion of Siegel's concerned only two logarithms, with severely restrict
ed (though completely non-trivial) quadratic algebraic " }{XPPEDIT 18
0 "beta[1],beta[2];" "6$&%%betaG6#\"\"\"&F$6#\"\"#" }{TEXT -1 143 ". W
hat a later triumph it was for Alan Baker to completely settle, not ju
st that question of Siegel, but the completely general version of it: \+
" }{TEXT 1098 1 "n" }{TEXT -1 27 " logarithms, and algebraic " }
{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 50 "s. I will come to that
in the later Baker section." }}{PARA 0 "" 0 "" {TEXT -1 57 " At th
e end of that same chapter Siegel also remarks: " }{TEXT 1099 119 "Ano
ther example showing the narrow limits of our knowledge on transcenden
tal numbers is the following one: Since e and " }{XPPEDIT 18 0 "Pi;" "
6#%#PiG" }{TEXT -1 1 " " }{TEXT 1100 25 "are both transcendental, " }
{TEXT 1101 8 "not both" }{TEXT 1102 9 " numbers " }{XPPEDIT 18 0 "e+Pi
;" "6#,&%\"eG\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 1103 3 "and" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "e*Pi;" "6#*&%\"eG\"\"\"%#PiGF%" }{TEXT
-1 1 " " }{TEXT 1104 17 "can be algebraic " }{TEXT -1 90 "[that, by th
e way, is simply a particular case of a completely general remark, nam
ely: if " }{XPPEDIT 18 0 "t[1],t[2];" "6$&%\"tG6#\"\"\"&F$6#\"\"#" }
{TEXT -1 31 " are both transcendental, then " }{XPPEDIT 18 0 "t[1]+t[2
];" "6#,&&%\"tG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 5 " and " }{XPPEDIT
18 0 "t[1]*t[2];" "6#*&&%\"tG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 26 " ca
nnot both be algebraic]" }{TEXT 1108 34 "; but we do not even know whe
ther " }{XPPEDIT 18 0 "e+Pi;" "6#,&%\"eG\"\"\"%#PiGF%" }{TEXT -1 1 " \+
" }{TEXT 1105 2 "or" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e*Pi;" "6#*&%\"eG
\"\"\"%#PiGF%" }{TEXT -1 1 " " }{TEXT 1106 15 "are irrational." }
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Kurt Mahler (190
3-1988)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
412 "Mahler contributed many, many beautiful results to transcendence \+
theory. He also formulated his famous A-, S-, T-, U-classification of \+
real and complex numbers, and proved that almost every real or complex
number is an S-number. I can't remember where I read that he used 'S'
as a tribute to Siegel (can anyone confirm that?), and simply used 'T
' and 'U' since they follow alphabetically! 'A', of course, is for " }
{TEXT 737 9 "algebraic" }{TEXT -1 160 ". Originally I had quite a bit \+
typed up of his classification - especially on its motivation - but in
a bad moment I removed it, as I could see no end in sight." }}{PARA
0 "" 0 "" {TEXT -1 305 " So, just some of Mahler's results, and I b
egin with one - from 1937 - which is often mentioned, possibly because
it is so eye-catching: the infinite decimal 0.1 2 3 4 5 6 7 8 9 10 11
12 13... is transcendental (can you see why it is certainly irrationa
l?). Actually Mahler's full theorem is that if f(" }{TEXT 738 1 "x" }
{TEXT -1 247 ") is any non-constant polynomial such that f(1), f(2), f
(3), ... are all natural numbers, then the decimal formed by concatena
tion of those values is transcendental. The result is true, in fact, n
ot just in base 10, but in any base 2, 3, 4, ... . " }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 125 "Msee := proc(f, r) local a, k; a[1]:=f(1):
\nfor k from 2 to r do a[k]:=a[k-1]*10^length(f(k))+f(k)\nod: print(a[
r]*`...`); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 28 "Thus, for example, choosing " }{XPPEDIT
18 0 "f(x) = x^2/2+x/2;" "6#/-%\"fG6#%\"xG,&*&F'\"\"#F*!\"\"\"\"\"*&F'
F,F*F+F," }{TEXT -1 36 ", we have that the infinite decimal:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Msee(x -> x^2/2 + x/2, 20);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "is transcendental by Mah
ler's theorem. Incidentally the decimal number 0.1 2 3 4 5 6 7 8 9 10 \+
11 12 13... is known as the Mahler-Champernowne number, with C's name \+
incorporated, since it was he who proved in a 1933 paper that the abov
e number is a " }{TEXT 739 6 "normal" }{TEXT -1 23 " number in the bas
e 10." }}{PARA 0 "" 0 "" {TEXT -1 112 " Many will have heard of the
famous Catalan problem (only recently solved): are 8 and 9 the only c
onsecutive " }{TEXT 726 6 "proper" }{TEXT -1 29 " powers? In other wor
ds are (" }{TEXT 727 1 "x" }{TEXT -1 2 ", " }{TEXT 728 1 "y" }{TEXT
-1 2 ", " }{TEXT 729 1 "m" }{TEXT -1 2 ", " }{TEXT 730 1 "n" }{TEXT
-1 64 ") = (3, 2, 2, 3) the only solutions of the Diophantine equation
" }{XPPEDIT 18 0 "x^m-y^n = 1;" "6#/,&)%\"xG%\"mG\"\"\")%\"yG%\"nG!\"
\"F(" }{TEXT -1 20 " in natural numbers " }{TEXT 720 1 "x" }{TEXT -1
2 ", " }{TEXT 721 2 "y," }{TEXT -1 1 " " }{TEXT 722 1 "m" }{TEXT -1 2
", " }{TEXT 723 1 "n" }{TEXT -1 7 ", with " }{TEXT 724 1 "m" }{TEXT
-1 5 " and " }{TEXT 725 1 "n" }{TEXT -1 162 " both greater than 1. Tha
t there could be at most a finite number of solutions is a minor conse
quence (which I leave as an exercise) of the much more substantial:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Mahler's
Theorem. Let " }{TEXT 731 1 "m" }{TEXT -1 5 " and " }{TEXT 732 1 "n"
}{TEXT -1 81 " be natural numbers - one of which is at least 2, the ot
her at least 3 - and let " }{TEXT 733 1 "a" }{TEXT -1 5 " and " }
{TEXT 734 1 "b" }{TEXT -1 61 " be any non-zero integers. Then the grea
test prime factor of " }{XPPEDIT 18 0 "f(x,y) = a*x^m+b*y^n;" "6#/-%\"
fG6$%\"xG%\"yG,&*&%\"aG\"\"\")F'%\"mGF,F,*&%\"bGF,)F(%\"nGF,F," }
{TEXT -1 26 " tends to infinity as max(" }{XPPEDIT 18 0 "abs(x),abs(y)
;" "6$-%$absG6#%\"xG-F$6#%\"yG" }{TEXT -1 68 ") tends to infinity, sub
ject to the restriction that gcd(x, y) = 1. " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Note that one does need the " }
{TEXT 735 34 "\"at least 2, the other at least 3\"" }{TEXT -1 17 ", as
the example " }{XPPEDIT 18 0 "x^2-2*y^2;" "6#,&*$%\"xG\"\"#\"\"\"*&F&
F'*$%\"yGF&F'!\"\"" }{TEXT -1 68 " shows (it is '1' infinitely often).
Note also the relevance of the " }{TEXT 745 13 "gcd(x, y) = 1" }
{TEXT -1 40 " restriction: if, for example, one took " }{XPPEDIT 18 0
"f(x,y) = x^3-2*y^3;" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"$\"\"\"*&\"\"#F,
*$F(F+F,!\"\"" }{TEXT -1 9 " and set " }{XPPEDIT 18 0 "x = 5.2^r;" "6#
/%\"xG)$\"#_!\"\"%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 4.2^r;
" "6#/%\"yG)$\"#U!\"\"%\"rG" }{TEXT -1 2 " (" }{TEXT 746 1 "r" }{TEXT
-1 41 " any natural number), then one would have" }}{PARA 258 "" 0 ""
{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y) = (5.2^r)^3-2*(3.2^r)^3;" "6#/-%
\"fG6$%\"xG%\"yG,&*$)$\"#_!\"\"%\"rG\"\"$\"\"\"*&\"\"#F1*$)$\"#KF.F/F0
F1F." }{TEXT -1 3 " = " }{XPPEDIT 18 0 "125.(2^(3*r))-54.(2^(3*r)) = 7
1.2^(3*r);" "6#/,&-$\"$D\"\"\"!6#)\"\"#*&\"\"$\"\"\"%\"rGF.F.-$\"#aF(6
#)F+*&F-F.F/F.!\"\")$\"$7(F6*&F-F.F/F." }{TEXT -1 2 ", " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "whose largest prime \+
factor (71) remains " }{TEXT 747 6 "static" }{TEXT -1 4 " as " }{TEXT
748 1 "r" }{TEXT -1 28 " is made increasingly large." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Mahler's proof of the l
atter theorem depended on his own " }{TEXT 736 1 "p" }{TEXT -1 41 "-ad
ic version of the Thue-Siegel theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 29 "Mahler's Theorem (1929). Let " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 39 " be any non-zero alg
ebraic number with " }{XPPEDIT 18 0 "abs(alpha) < 1;" "6#2-%$absG6#%&a
lphaG\"\"\"" }{TEXT -1 10 ", and let " }{XPPEDIT 18 0 "omega;" "6#%&om
egaG" }{TEXT -1 48 " be any real algebraic number of degree 2, then "
}{XPPEDIT 18 0 "sum([n*omega]*alpha^n,n = 1 .. infinity);" "6#-%$sumG6
$*&7#*&%\"nG\"\"\"%&omegaGF*F*)%&alphaGF)F*/F);F*%)infinityG" }{TEXT
-1 27 " is transcendental (where [" }{TEXT 719 1 "x" }{TEXT -1 31 "] d
enotes the integral part of " }{TEXT 718 1 "x" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 29 "Mahler's Theorem (1929). Let " }{XPPEDIT
18 0 "f(z) = sum(z^(2^n),n = 0 .. infinity);" "6#/-%\"fG6#%\"zG-%$sumG
6$)F')\"\"#%\"nG/F.;\"\"!%)infinityG" }{TEXT -1 4 " (= " }{XPPEDIT 18
0 "1+z^2+z^4+z^8+`...`;" "6#,,\"\"\"F$*$%\"zG\"\"#F$*$F&\"\"%F$*$F&\"
\")F$%$...GF$" }{TEXT -1 12 " ), and let " }{XPPEDIT 18 0 "alpha;" "6#
%&alphaG" }{TEXT -1 17 " be any non-zero " }}{PARA 0 "" 0 "" {TEXT -1
22 "algebraic number with " }{XPPEDIT 18 0 "abs(alpha) < 1;" "6#2-%$ab
sG6#%&alphaG\"\"\"" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "f(alpha);" "
6#-%\"fG6#%&alphaG" }{TEXT -1 20 " is transcendental. " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "There is a very fin
e exposition of those two latter results, with many extensions, in a p
aper by Loxton and van der Poorten in the Baker & Masser 1977 " }
{TEXT 744 11 "Proceedings" }{TEXT -1 2 ". " }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 29 "Theodor Schneider (1911-1988)" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 300 "I once heard Schneider give a \+
seminar talk; it was at Imperial College (London), in the late-60s/ear
ly-70s. He was introduced by the legendary K. F. Roth, who was then at
IC. Roth introduced Schneider as being someone who needed no introduc
tion, who had done this great thing and that great thing in " }{TEXT
626 13 "transcendence" }{TEXT -1 84 ". His talk - which I can only vag
uely recollect - was roughly about this: suppose f(" }{TEXT 622 1 "x"
}{TEXT -1 2 ", " }{TEXT 623 1 "y" }{TEXT -1 78 ") is a 2-variable poly
nomial with rational coefficients such that the curve f(" }{TEXT 624
1 "x" }{TEXT -1 2 ", " }{TEXT 625 1 "y" }{TEXT -1 214 ") = 0 has a poi
nt on it with both co-ordinates algebraic numbers of degree 5, then - \+
subject to certain complicated conditions - that curve has a rational \+
point on it (i.e. both co-ordinates are rarional numbers). " }}{PARA
0 "" 0 "" {TEXT -1 250 " There were many 'names' at the talk, at th
e end of which there was a deathly silence when Roth asked if anyone h
ad any questions to ask... (Everyone knows that silence!) Some very em
inent listener (whose name shall remain a secret!) blurted out: " }
{TEXT 627 64 "and has your result got any other applications in transc
endence?" }{TEXT -1 59 " Poor Schneider looked absolutely aghast, and \+
said that he " }{TEXT 628 39 "hadn't been talking about transcendence
" }{TEXT -1 104 "... Blushes all round, and Roth jumped in to relieve \+
the situation by asking some sensible questions... " }}{PARA 0 "" 0 "
" {TEXT -1 321 " Finally, the talk was in one of those old rooms, w
ith old school desks (is it still like this?), so that one was sitting
beside just one other person. The person beside me and myself introdu
ced each other; he told me - in an American accent - that he was 'Paul
Cohen', and I wondered - without asking him - if he was " }{TEXT 629
4 "the " }{TEXT -1 172 "Paul Cohen (1966 Fields Medallist, famed for h
is work on the independence of the Continuum Hypothesis and Axiom of C
hoice etc). A few minutes later I said to someone (RB): " }{TEXT 630
86 "do you see that guy over there, he told me he was Paul Cohen; do y
ou suppose he is...?" }{TEXT -1 38 " RB confirmed that he was. Happy d
ays." }}{PARA 0 "" 0 "" {TEXT -1 109 " Schneider features in the Hi
lbert's seventh problem section later, and here, as with Siegel, I giv
e only " }{TEXT 1179 18 "some small flavour" }{TEXT -1 55 " of the sor
t of contribution made by this great figure." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 919 26 "Schneider's (1937) Theorem" }
{TEXT -1 55 " (recall Siegel's earlier 1932 related theorem). Let P("
}{TEXT 920 1 "z" }{TEXT -1 60 ") be the Weirstrass elliptic function w
ith algebraic number " }{TEXT 921 10 "invariants" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "g[2];" "6#&%\"gG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT
18 0 "g[3];" "6#&%\"gG6#\"\"$" }{TEXT -1 9 " in (the " }{TEXT 954 8 "s
tandard" }{TEXT -1 21 " equation relating P(" }{TEXT 952 1 "z" }{TEXT
-1 9 ") and P'(" }{TEXT 953 1 "z" }{TEXT -1 2 "))" }}{PARA 258 "" 0 "
" {TEXT -1 0 "" }{XPPEDIT 18 0 "diff(P(z),z)^2 = 4*P(z)^3-g[2]*P(z)-g[
3];" "6#/*$-%%diffG6$-%\"PG6#%\"zGF+\"\"#,(*&\"\"%\"\"\"*$-F)6#F+\"\"$
F0F0*&&%\"gG6#F,F0-F)6#F+F0!\"\"&F76#F4F;" }{TEXT -1 2 "; " }}{PARA 0
"" 0 "" {TEXT -1 5 "then " }{TEXT 924 3 "any" }{TEXT -1 24 " (non-zero
) period of P(" }{TEXT 922 1 "z" }{TEXT -1 27 ") is transcendental. Al
so, " }{XPPEDIT 18 0 "P(alpha);" "6#-%\"PG6#%&alphaG" }{TEXT -1 37 " i
s transcendental for any algebraic " }{XPPEDIT 18 0 "alpha,alpha <> 0;
" "6$%&alphaG0F#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 25 "For the next, recall the " }{TEXT
925 2 "j(" }{TEXT -1 1 "z" }{TEXT 926 43 ")-modular function from an e
arlier section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT 927 26 "Schneider's (1937) Theorem" }{TEXT -1 7 ". Let " }
{XPPEDIT 18 0 "alpha = alpha[1]+i*alpha[2];" "6#/%&alphaG,&&F$6#\"\"\"
F(*&%\"iGF(&F$6#\"\"#F(F(" }{TEXT -1 20 " be algebraic, with " }
{XPPEDIT 18 0 "alpha[1],alpha[2];" "6$&%&alphaG6#\"\"\"&F$6#\"\"#" }
{TEXT -1 10 " real and " }{XPPEDIT 18 0 "0 < alpha[2];" "6#2\"\"!&%&al
phaG6#\"\"#" }{TEXT -1 7 "; then " }{XPPEDIT 18 0 "j(alpha);" "6#-%\"j
G6#%&alphaG" }{TEXT -1 28 " is transcendental whenever " }{XPPEDIT 18
0 "alpha;" "6#%&alphaG" }{TEXT -1 14 " is of degree " }{TEXT 928 7 "gr
eater" }{TEXT -1 8 " than 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }{TEXT 1178 27 "Schneider's ellipse Theorem" }
{TEXT -1 19 ". Any ellipse with " }{TEXT 1180 9 "algebraic" }{TEXT -1
56 " major and minor axes has transcendental circumference. " }}{PARA
0 "" 0 "" {TEXT -1 23 " In other words, if " }{TEXT 1181 1 "a" }
{TEXT -1 5 " and " }{TEXT 1182 1 "b" }{TEXT -1 74 " are algebraic numb
ers, then the circumference of the curve with equation " }{XPPEDIT 18
0 "x^2/(a^2)+y^2/(b^2) = 1;" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%
\"yGF'*$%\"bGF'F*F+F+" }{TEXT -1 67 ", has transcendental circumferenc
e. Note that the transcendence of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 40 " is a minor consequence of this theorem." }}}{SECT 1
{PARA 4 "" 0 "" {TEXT -1 29 "Klaus Friedrich Roth (1925- )" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Roth's Theorem (1
955, settling Siegel's conjecture). Let " }{XPPEDIT 18 0 "alpha;" "6#%
&alphaG" }{TEXT -1 40 " be any real algebraic number of degree " }
{TEXT 399 1 "n" }{TEXT -1 2 " (" }{TEXT 403 8 "at least" }{TEXT -1 13
" 3), and let " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 61
" be any positive constant, then there is a positive constant " }
{XPPEDIT 18 0 "c = c(alpha,epsilon);" "6#/%\"cG-F$6$%&alphaG%(epsilonG
" }{TEXT -1 20 " (i.e. the value of " }{TEXT 400 1 "c" }{TEXT -1 9 " d
epends " }{TEXT 401 4 "only" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "alpha;
" "6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "epsilon;" "6#%(epsi
lonG" }{TEXT -1 11 ") such that" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "ab
s(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%\"qG!\"\"F," }
{TEXT -1 1 " " }{TEXT 402 1 ">" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c(alph
a)/(q^(2+epsilon));" "6#*&-%\"cG6#%&alphaG\"\"\")%\"qG,&\"\"#F(%(epsil
onGF(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 94 "That was not how Roth himself expressed his resul
t; he expressed it (equivalently) as follows:" }}{PARA 0 "" 0 ""
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 60 " \+
be any real irrational algebraic number. If the inequality " }
{XPPEDIT 18 0 "abs(alpha-p/q) < 1/(q^kappa);" "6#2-%$absG6#,&%&alphaG
\"\"\"*&%\"pGF)%\"qG!\"\"F-*&F)F))F,%&kappaGF-" }{TEXT -1 8 " (where \+
" }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 64 " is a constant) h
as an infinite number of solutions in integers " }{TEXT 887 1 "p" }
{TEXT -1 5 " and " }{TEXT 888 1 "q" }{TEXT -1 7 " (with " }{XPPEDIT
18 0 "0 < q;" "6#2\"\"!%\"qG" }{TEXT -1 7 ") then " }{XPPEDIT 18 0 "ka
ppa <= 2;" "6#1%&kappaG\"\"#" }{TEXT -1 2 ". " }}{PARA 258 "" 0 ""
{TEXT -1 14 "______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 173 "In his address at the 1958 International Congr
ess of Mathematicians - held in Edinburgh, at which Roth was awarded o
ne of the two Fields Medals - Harold Davenport commented:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 429 0 "" }{TEXT 397 16
"The achievement " }{TEXT -1 9 "[of Roth]" }{TEXT 433 1 " " }{TEXT
398 83 "is one that speaks for itself: it closes a chapter, and a new \+
chapter is now opened" }{TEXT 432 1 " " }{TEXT -1 260 "[although Daven
port didn't enlarge, he must have intended the problem of simultaneous
approximations, of which Roth himself spoke in his own address to the
Congress, and which was later settled - partially in 1967, and comple
tely in 1970 - by Wolfgang Schmidt]" }{TEXT 431 185 ". Roth's theorem \+
settles a question which is both of a fundamental nature and of extrem
e difficulty. It will stand as a landmark in mathematics for as long a
s mathematics is cultivated." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "In the 1956 Preface to his " }
{TEXT 885 42 "Introduction to Diophantine Approximation " }{TEXT -1
15 "Cassels wrote: " }{TEXT 886 52 "' like to express my gratitude to \+
... and Dr. K. F. " }{TEXT 889 4 "Roth" }{TEXT 890 25 " put a manuscri
pt of his " }{TEXT 891 25 "revolutionary improvement" }{TEXT 892 63 " \+
of the Thue-Siegel Theorem at my disposal before publication.'" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 706 "Personal
anecdote. When I was a student in London, I once asked Roth (in his o
ffice at Imperial College) what were the circumstances in which he pro
ved his famous 1955 result. Roth told me that when he worked with Dave
nport in University College London in the early 50's, Davenport had a \+
practice of inviting colleagues to read up on some difficult piece of \+
work, and then explain it in a seminar talk. Davenport asked him to re
ad the Thue-Siegel result. He read it, understood it, explained it to \+
everyone, and then (after all that effort) decided to give himself one
year (Roth's standard practice) to solve Siegel's conjectured improve
ment. His year was almost up, he was just about to give up, when... "
}}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 337 "On
ce I attended a seminar talk at my college (Royal Holloway) by the emi
nent combinatorialist Richard Rado (incidentally, his son, Peter, was \+
also studying at RHC at that time). Afterwards I saw Rado sitting alon
e in the library, and I approached him to express my appreciation of h
is talk. He asked about my interests, and, when I said " }{TEXT 427
13 "Number Theory" }{TEXT -1 1 "," }{TEXT 428 1 " " }{TEXT -1 116 "he \+
expressed the view that Roth's rational approximations theorem was the
greatest result in all of Number Theory. " }}{PARA 0 "" 0 "" {TEXT
-1 76 " Incidentally, Roth gets a passing mention in Sylvia Nasar's
bestseller, " }{TEXT 430 17 "A Beautiful Mind " }{TEXT -1 341 "(I'm s
ure that most of my readers will have seen the film of the same name -
and possibly have read Nasar's book? - with Russell Crowe playing the
part of John Nash). Considering the background to the awarding of the
two 1958 Fields Medals - apparently Nash was one of the thirty-six no
minees - she writes (p.226 of my hardback 1st edition) " }{TEXT 434 4
"Roth" }{TEXT 517 14 " was a shoo-in" }{TEXT -1 18 " [for the Medal]; \+
" }{TEXT 435 92 "he had solved a fundamental problem in number theory \+
that the most senior committee member, " }{TEXT 559 18 "Carl Ludwig Si
egel" }{TEXT 560 36 ", had worked on early in his career." }{TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "A
t the end of his Fields Medal acceptance talk, at the Edinburgh 1958 I
nternational Congress of Mathematicians, Roth spoke thus: " }{TEXT
941 249 "One outstanding problem is to obtain a theorem analogous to o
urs concerning simulatneous approximations to two or more algebraic nu
mbers by rational numbers of the same denominator. In the case of simu
ltaneous approximation to two algebraic numbers " }{XPPEDIT 18 0 "alph
a[1],alpha[2];" "6$&%&alphaG6#\"\"\"&F$6#\"\"#" }{TEXT -1 1 " " }
{TEXT 942 86 "(subject to a suitable independence condition), one woul
d expect that the inequalities" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(alpha[1]-p[1]/q) < 1/(q^kappa);"
"6#2-%$absG6#,&&%&alphaG6#\"\"\"F+*&&%\"pG6#F+F+%\"qG!\"\"F1*&F+F+)F0%
&kappaGF1" }{TEXT -1 9 ", " }{XPPEDIT 18 0 "abs(alpha[2]-p[2]/q
) < 1/(q^kappa);" "6#2-%$absG6#,&&%&alphaG6#\"\"#\"\"\"*&&%\"pG6#F+F,%
\"qG!\"\"F2*&F,F,)F1%&kappaGF2" }{TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 943 45 "to have at most a finite
number of solutions " }{TEXT -1 13 "[in integers " }{XPPEDIT 18 0 "p[
1],p[2],q;" "6%&%\"pG6#\"\"\"&F$6#\"\"#%\"qG" }{TEXT -1 16 ", with pos
itive " }{TEXT 944 1 "q" }{TEXT -1 2 "] " }{TEXT 945 8 "for any " }
{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 1 " " }{TEXT 946 1 ">" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }
{TEXT -1 2 ". " }{TEXT 947 171 "But practically nothing is known in th
is direction. A complete solution to the problem of simultaneous appro
ximations could lead to the complete solution of many others..." }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " What Roth had in mind
by " }{TEXT 948 47 "' subject to a suitable independence condition'"
}{TEXT -1 27 " is that the three numbers " }{XPPEDIT 18 0 "1,alpha[1],
alpha[2];" "6%\"\"\"&%&alphaG6#F#&F%6#\"\"#" }{TEXT -1 104 " should be
linearly independent over the rationals, otherwise it is possible to \+
construct examples with " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT
-1 1 " " }{TEXT 949 1 ">" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/2;" "6#*&
\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 78 " for which the above simultaneous
inequalities have infinitely many solutions." }}}{SECT 1 {PARA 4 ""
0 "" {TEXT -1 25 "Wolfgang Schmidt (1933- )" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Schmidt's Theorem 1 (1967). Let
" }{XPPEDIT 18 0 "alpha[1];" "6#&%&alphaG6#\"\"\"" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "alpha[2];" "6#&%&alphaG6#\"\"#" }{TEXT -1 37 " be re
al algebraic numbers such that " }{XPPEDIT 18 0 "1,alpha[1],alpha[2];
" "6%\"\"\"&%&alphaG6#F#&F%6#\"\"#" }{TEXT -1 61 " are linearly indepe
ndent over the rational numbers, and let " }{XPPEDIT 18 0 "epsilon;" "
6#%(epsilonG" }{TEXT -1 84 " > 0. Then there are only a finite number \+
of (simultaneous) rational approximations " }{XPPEDIT 18 0 "p[1]/q;" "
6#*&&%\"pG6#\"\"\"F'%\"qG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[
2]/q;" "6#*&&%\"pG6#\"\"#\"\"\"%\"qG!\"\"" }{TEXT -1 10 " such that" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "abs(
alpha[1]-p[1]/q) < 1/(q^(3/2+epsilon));" "6#2-%$absG6#,&&%&alphaG6#\"
\"\"F+*&&%\"pG6#F+F+%\"qG!\"\"F1*&F+F+)F0,&*&\"\"$F+\"\"#F1F+%(epsilon
GF+F1" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "abs(alpha[2]-p[2]/q) < \+
1/(q^(3/2+epsilon));" "6#2-%$absG6#,&&%&alphaG6#\"\"#\"\"\"*&&%\"pG6#F
+F,%\"qG!\"\"F2*&F,F,)F1,&*&\"\"$F,F+F2F,%(epsilonGF,F2" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "Schmidt's Theorem 2 (1970). Let " }
{XPPEDIT 18 0 "alpha[1],`...`,alpha[n];" "6%&%&alphaG6#\"\"\"%$...G&F$
6#%\"nG" }{TEXT -1 37 " be real algebraic numbers such that " }
{XPPEDIT 18 0 "1,alpha[1],`...`,alpha[n];" "6&\"\"\"&%&alphaG6#F#%$...
G&F%6#%\"nG" }{TEXT -1 63 " are linearly independent over the rational
numbers Q, and let " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT
-1 82 " > 0. Then there are only a finite number of simultaneous ratio
nal approximations " }{XPPEDIT 18 0 "p[1]/q,`...`,p[n]/q;" "6%*&&%\"pG
6#\"\"\"F'%\"qG!\"\"%$...G*&&F%6#%\"nGF'F(F)" }{TEXT -1 10 " such that
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "a
bs(alpha[1]-p[1]/q) < 1/(q^(1+1/n+epsilon));" "6#2-%$absG6#,&&%&alphaG
6#\"\"\"F+*&&%\"pG6#F+F+%\"qG!\"\"F1*&F+F+)F0,(F+F+*&F+F+%\"nGF1F+%(ep
silonGF+F1" }{TEXT -1 9 ", ... , " }{XPPEDIT 18 0 "abs(alpha[n]-p[n]/
q) < 1/(q^(1+1/n+epsilon));" "6#2-%$absG6#,&&%&alphaG6#%\"nG\"\"\"*&&%
\"pG6#F+F,%\"qG!\"\"F2*&F,F,)F1,(F,F,*&F,F,F+F2F,%(epsilonGF,F2" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 301 "I regret
I have omitted Mahler's famous A-, S-, T-, U- classification of real \+
and complex numbers, which I briefly mentioned in the Mahler section e
arlier. Suffice it to remark that algebraic numbers are of type A, whi
le transcendental numbers fall into the other three categories: roughl
y those that " }{TEXT 929 27 "cannot be well-approximated" }{TEXT -1
60 " by algebraic numbers (those are the S-numbers), those that " }
{TEXT 930 24 "can be well-approximated" }{TEXT -1 113 " by algebraic n
umbers (those are the U-numbers, which in turn are further classified \+
according to their integral " }{TEXT 933 6 "degree" }{TEXT -1 133 ", a
nd here the Liouville numbers are an extreme example: those of degree \+
1), and finally those as it were in between (the T-numbers)." }}{PARA
0 "" 0 "" {TEXT -1 250 " Mahler himself proved that almost all (in \+
the exact sense of Lebesgue measure theory) real or complex numbers ar
e of S-type. There was then a long-standing conjecture of Mahler's - d
ating from 1932 - that almost all real numbers are S-numbers of " }
{TEXT 931 5 "type " }{TEXT -1 51 "1, and almost all complex numbers ar
e S-numbers of " }{TEXT 932 4 "type" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1
/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 106 ". That was settled in 196
5 by Sprindzuk. In 1953 LeVeque proved the existence of U-numbers of e
ach degree." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 ""
{TEXT -1 73 "Schmidt's fundamental contribution, in this connection, w
as to prove the " }{TEXT 934 9 "existence" }{TEXT -1 69 " of T-numbers
. I quote from the Introduction to Schmidt's 1970 paper " }{TEXT 935
1 "T" }{TEXT -1 44 "-NUMBERS DO EXIST (from a 1968 conference): " }
{TEXT 936 323 "K. Mahler in 1932 divided the real transcendental numbe
rs into three classes, and called numbers in these classes S-numbers, \+
T-numbers and U-numbers. But while the existence of S-numbers and of U
-numbers is easy to see, the existence of T-numbers was left open. It \+
is the purpose of the present paper to prove the following" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 937 29 "T
heorem 1. T-numbers do exist" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 938 54 "The proof wi
ll be via T*-numbers introduced by Koksma " }{TEXT -1 8 "[1939]. " }
{TEXT 939 142 "We shall make essential use of a recent theorem of Wirs
ing about approximations to an algebraic number by algebraic numbers o
f a given degree." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Hilbert's \+
seventh problem. Gelfond, Schneider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 65 "General background: Hilbert's famous (23
) MATHEMATICAL PROBLEMS (" }{TEXT 558 86 "LECTURE DELIVERED BEFORE THE
INTERNATIONAL CONGRESS OF MATHEMATICIANS AT PARIS IN 1900" }{TEXT -1
2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 543 25 "Hilbert's seventh problem" }{TEXT -1 44 " (after a pr
eamble) asked for a proof that (" }{TEXT 637 3 "any" }{TEXT -1 11 " va
lue of) " }{XPPEDIT 18 0 "alpha^beta;" "6#)%&alphaG%%betaG" }{TEXT -1
16 " (for algebraic " }{XPPEDIT 18 0 "alpha <> 0,1;" "6$0%&alphaG\"\"!
\"\"\"" }{TEXT -1 6 ", and " }{TEXT 557 10 "irrational" }{TEXT -1 11 "
algebraic " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 8 "), e.g. \+
" }{XPPEDIT 18 0 "2^sqrt(2);" "6#)\"\"#-%%sqrtG6#F$" }{TEXT -1 4 " or \+
" }{XPPEDIT 18 0 "e^Pi = i^(-2*i);" "6#/)%\"eG%#PiG)%\"iG,$*&\"\"#\"\"
\"F(F,!\"\"" }{TEXT -1 71 ", is transcendental (or at least an irratio
nal number). Hilbert wrote: " }{TEXT 556 188 "\" It is certain that th
e solution of these and similar problems must lead us to entirely new \+
methods and to a new insight into the nature of special irrational and
transcendental numbers.\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
30 "evalc(sqrt(-1)^(-2*sqrt(-1)));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 40 "evalc((-1)^(-sqrt(-1))); # alternatively" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalc(I^(-2*I)); # alternatively" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Although Hilbert didn't specif
ically refer to (what I earlier called) Euler's surmise, he must have \+
had it in mind..." }}{PARA 258 "" 0 "" {TEXT -1 10 "__________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "From Cons
tance Reid's biography of Hilbert (p.164) I quote: " }{TEXT 546 0 "" }
{TEXT 547 423 "\"Siegel came to G\366ttingen as a student in 1919... h
e was always to remember a lecture on number theory which he heard fro
m Hilbert at this time. Hilbert wanted to give his listeners examples \+
of the characteristic problems of the theory of numbers which seem at \+
first glance so very simple but turn out to be incredibly difficult to
solve. He mentioned Riemann's hypothesis, Fermat's [Last] theorem, an
d the transcendence of" }{TEXT 631 1 " " }{XPPEDIT 18 0 "2^sqrt(2);" "
6#)\"\"#-%%sqrtG6#F$" }{TEXT 548 424 "as examples of this type of prob
lem. Then he went on to say that there had recently been much progress
on Riemann's hypothesis and he was very hopeful that he would live to
see it proved. Fermat's problem had been around for a very long time \+
and apparently demanded entirely new methods for its solution - perhap
s the youngest members of his audience would live to see it solved. Bu
t as for establishing the transcendence of" }{TEXT 632 1 " " }
{XPPEDIT 18 0 "2^sqrt(2);" "6#)\"\"#-%%sqrtG6#F$" }{TEXT -1 0 "" }
{TEXT 549 58 "no one present in the lecture hall would live to see tha
t!" }}{PARA 0 "" 0 "" {TEXT 633 266 " The first two problems which \+
Hilbert mentioned are still unsolved [JC comment. As everyone knows, F
ermat's Last Theorem has been settled by Andrew Wiles]. But less than \+
ten years later a young Russian mathematician named Gelfond establishe
d the transcendence of " }{XPPEDIT 18 0 "2^sqrt(-2);" "6#)\"\"#-%%sqrt
G6#,$F$!\"\"" }{TEXT -1 0 "" }{TEXT 552 0 "" }{TEXT 553 1 " " }{TEXT
634 2 ". " }{TEXT -1 0 "" }{TEXT 555 94 "Utilising this work, Siegel h
imself was shortly able to establish the desired transcendence of" }
{TEXT 635 1 " " }{XPPEDIT 18 0 "2^sqrt(2)" "6#)\"\"#-%%sqrtG6#F$" }
{TEXT 554 1 "." }}{PARA 0 "" 0 "" {TEXT 940 252 " Siegel wrote to H
ilbert about the proof. He reminded him of what he had said in his 192
0 lecture and emphasised that the important work was that of Gelfond. \+
Hilbert was frequently criticized for \"acting as if everything had be
en done in G\366ttingen.\"" }{TEXT 550 0 "" }{TEXT 551 309 " Now he re
sponded with enthusiastic delight to Siegel's letter, but the made no \+
mention of the young Russian's contribution. He wanted only to publish
Siegel's solution. Siegel refused, certain that Gelfond himself would
eventually solve this problem too. Hilbert immediately lost all inter
est in the matter.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 213 "As it happened, Gelfond didn't find a solution himself
, and another Russian mathematician - Kuzmin - published a solution th
e following year. In 1933, K. Boehle, using Gelfond's method did manag
e to prove that if " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 24 "
is algebraic of degree " }{TEXT 711 1 "n" }{TEXT -1 48 " (at least 3)
, then at least one of the numbers " }{XPPEDIT 18 0 "alpha^beta,alpha^
(beta^2),`...`,alpha^(beta^(n-1));" "6&)%&alphaG%%betaG)F$*$F%\"\"#%$.
..G)F$)F%,&%\"nG\"\"\"F.!\"\"" }{TEXT -1 338 " is transcendental, and \+
then in 1934, with some justice, Gelfond himself published a complete \+
solution (April 1st 1934?) to the entire seventh problem of Hilbert. W
ithin six weeks, a PhD student of Siegel's - the great Schneider - fou
nd another independent solution, which he submitted on May 28th 1934, \+
and was published later that year." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "Gelfond's (1929) Theorem. Let " }
{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 26 " be any algebraic nu
mber (" }{XPPEDIT 18 0 "alpha <> 0,1;" "6$0%&alphaG\"\"!\"\"\"" }
{TEXT -1 10 ") and let " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1
104 " be any imaginary quadratic (i.e. a solution of a quadratic equat
ion with negative discriminant), then (" }{TEXT 636 3 "any" }{TEXT -1
11 " value of) " }{XPPEDIT 18 0 "alpha^beta;" "6#)%&alphaG%%betaG" }
{TEXT -1 19 " is transcendental." }}{PARA 0 "" 0 "" {TEXT -1 4 " "
}}{PARA 0 "" 0 "" {TEXT -1 310 "I believe it is possible for a novice \+
to follow at least the gist of Gelfond's 1929 proof (if I cut some big
corners, and hand wave just a little). It is helpful, I believe, if o
ne follows the proof of the following very beautiful result (though An
alysis folk would probably regard it as being rather trivial):" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Theorem (
Polya 1916). Let f(" }{TEXT 655 1 "z" }{TEXT -1 27 ") be an entire fun
ction of " }{TEXT 654 1 "z" }{TEXT -1 63 " such that f(0), f(1), f(2),
... are all integers, and suppose " }}{PARA 0 "" 0 "" {TEXT -1 2 "f(
" }{TEXT 656 1 "z" }{TEXT -1 8 ") grows " }{TEXT 657 6 "slower" }
{TEXT -1 6 " than " }{XPPEDIT 18 0 "2^z;" "6#)\"\"#%\"zG" }{TEXT -1
20 " (in the sense that " }{XPPEDIT 18 0 "abs(f(z)) <= C*2^(A*abs(z));
" "6#1-%$absG6#-%\"fG6#%\"zG*&%\"CG\"\"\")\"\"#*&%\"AGF--F%6#F*F-F-" }
{TEXT -1 8 ", where " }{TEXT 658 2 "A " }{TEXT -1 3 "and" }{TEXT 660
2 " C" }{TEXT -1 24 " and are constants, and " }{XPPEDIT 18 0 "A < 1;
" "6#2%\"AG\"\"\"" }{TEXT -1 10 "), then f(" }{TEXT 659 1 "z" }{TEXT
-1 65 ") is a polynomial [it's a trivial exercise that the polynomial \+
f(" }{TEXT 686 1 "z" }{TEXT -1 35 ") must have rational coefficients].
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Recal
l that an entire function f(" }{TEXT 662 1 "z" }{TEXT -1 26 ") of the \+
complex variable " }{TEXT 661 1 "z" }{TEXT -1 33 ", is one that is def
ined for all " }{TEXT 663 1 "z" }{TEXT -1 9 ", and is " }{TEXT 687 14
"differentiable" }{TEXT -1 1 " " }{TEXT 688 10 "everywhere" }{TEXT -1
6 " (f '(" }{TEXT 664 1 "z" }{TEXT -1 30 ") exists). Simple examples a
re" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 665 3 "al
l" }{TEXT -1 33 " polynomials (of finite degree) \n" }{XPPEDIT 18 0 "z
*(z+1)/2 = z^2/2+z/2;" "6#/*(%\"zG\"\"\",&F%F&F&F&F&\"\"#!\"\",&*&F%F(
F(F)F&*&F%F&F(F)F&" }{TEXT -1 20 ", is an integer for " }{TEXT 666 1 "
z" }{TEXT -1 20 " = 0, 1, 2, 3, ...\n\n" }{XPPEDIT 18 0 "z*(z+1)/3 = z
^2/3+z/3;" "6#/*(%\"zG\"\"\",&F%F&F&F&F&\"\"$!\"\",&*&F%\"\"#F(F)F&*&F
%F&F(F)F&" }{TEXT -1 38 ", is an integer for some, but not all " }
{TEXT 667 1 "z" }{TEXT -1 19 " = 0, 1, 2, 3, ... " }}{PARA 15 "" 0 ""
{XPPEDIT 18 0 "sin(z),cos(z),e^z,2^z,3^z,(2^z+3^z)/2,`...`;" "6)-%$sin
G6#%\"zG-%$cosG6#F&)%\"eGF&)\"\"#F&)\"\"$F&*&,&)F-F&\"\"\")F/F&F3F3F-!
\"\"%$...G" }{TEXT -1 68 " \nsin(0) = 0, but none of sin(1), sin(2), s
in(3), ... is an integer\n" }}{PARA 0 "" 0 "" {TEXT -1 61 "(sketch of)
Proof of Polya's theorem. The key is to expand f(" }{TEXT 669 1 "z" }
{TEXT -1 7 ") as a " }{TEXT 668 27 "Newton interpolation series" }
{TEXT -1 58 " (which is different from the usual Taylor series) at the
" }{TEXT 670 23 "points of interpolation" }{TEXT -1 18 " 0, 1, 2, 3, \+
... :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT
18 0 "f(z) = A[0]+A[1]*(z-1)+A[2]*z*(z-1)+`...`+A[n]*z*(z-1)*`...`*(z-
(n-1))+`...`;" "6#/-%\"fG6#%\"zG,.&%\"AG6#\"\"!\"\"\"*&&F*6#F-F-,&F'F-
F-!\"\"F-F-*(&F*6#\"\"#F-F'F-,&F'F-F-F2F-F-%$...GF-*,&F*6#%\"nGF-F'F-,
&F'F-F-F2F-F8F-,&F'F-,&F " 0 "" {MPLTEXT 1 0 25 "ev
alc((-1)^(-sqrt(-1))); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "
evalc(((-1)^(-sqrt(-1)))^(3 + 4*sqrt(-1)));" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 45 "evalc(((-1)^(-sqrt(-1)))^(-4 - 5*sqrt(-1))); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Arrange the Gaussian integers -
namely all complex numbers of the form " }{XPPEDIT 18 0 "a+i*b;" "6#,
&%\"aG\"\"\"*&%\"iGF%%\"bGF%F%" }{TEXT -1 8 ", where " }{TEXT 695 1 "a
" }{TEXT -1 5 " and " }{TEXT 696 1 "b" }{TEXT -1 35 " vary over the (o
rdinary) integers " }{XPPEDIT 18 0 "`...`,-3,-2,-1,0,1,2,3,`...`;" "6+
%$...G,$\"\"$!\"\",$\"\"#F&,$\"\"\"F&\"\"!F*F(F%F#" }{TEXT -1 18 ") - \+
in a sequence " }{XPPEDIT 18 0 "z[0],z[1],z[2],`...`;" "6&&%\"zG6#\"\"
!&F$6#\"\"\"&F$6#\"\"#%$...G" }{TEXT -1 14 " according to " }{TEXT
693 18 "increasing modulus" }{TEXT -1 46 ", and where several have the
same modulus, by " }{TEXT 694 19 "increasing argument" }{TEXT -1 40 "
, and then expand the (entire) function " }{XPPEDIT 18 0 "e^(Pi*z)" "6
#)%\"eG*&%#PiG\"\"\"%\"zGF'" }{TEXT -1 73 " as a Newton interpolation \+
series with the above points of interpolation:" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "e^(Pi*z) = A[0]+A[1]*
(z-z[0])*(z-z[1])+A[2]*(z-z[0])*(z-z[1])+`...`+A[n]*(z-z[0])*(z-z[1])*
`...`*(z-z[n-1])+`...`;" "6#/)%\"eG*&%#PiG\"\"\"%\"zGF(,.&%\"AG6#\"\"!
F(*(&F,6#F(F(,&F)F(&F)6#F.!\"\"F(,&F)F(&F)6#F(F5F(F(*(&F,6#\"\"#F(,&F)
F(&F)6#F.F5F(,&F)F(&F)6#F(F5F(F(%$...GF(*,&F,6#%\"nGF(,&F)F(&F)6#F.F5F
(,&F)F(&F)6#F(F5F(FCF(,&F)F(&F)6#,&FGF(F(F5F5F(F(FCF(" }}{PARA 258 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0
"C[n];" "6#&%\"CG6#%\"nG" }{TEXT -1 58 " be a circle centered at the o
rigin containing the points " }{XPPEDIT 18 0 "z[0],z[1],`...`,z[n];" "
6&&%\"zG6#\"\"!&F$6#\"\"\"%$...G&F$6#%\"nG" }{TEXT -1 6 ", then" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "A[n] \+
= 1/(2*Pi*i);" "6#/&%\"AG6#%\"nG*&\"\"\"F)*(\"\"#F)%#PiGF)%\"iGF)!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(f(z)/((z-z[0])*(z-z[1])*`...`*(z
-z[n])),z);" "6#-%$intG6$*&-%\"fG6#%\"zG\"\"\"**,&F*F+&F*6#\"\"!!\"\"F
+,&F*F+&F*6#F+F1F+%$...GF+,&F*F+&F*6#%\"nGF1F+F1F*" }}{PARA 258 "" 0 "
" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "f(
z[0])/((z[0]-z[1])*(z[0]-z[2])*`...`*(z[0]-z[n]));" "6#*&-%\"fG6#&%\"z
G6#\"\"!\"\"\"**,&&F(6#F*F+&F(6#F+!\"\"F+,&&F(6#F*F+&F(6#\"\"#F2F+%$..
.GF+,&&F(6#F*F+&F(6#%\"nGF2F+F2" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "f(z
[1])/((z[1]-z[0])*(z[1]-z[2])*`...`*(z[1]-z[n]));" "6#*&-%\"fG6#&%\"zG
6#\"\"\"F***,&&F(6#F*F*&F(6#\"\"!!\"\"F*,&&F(6#F*F*&F(6#\"\"#F2F*%$...
GF*,&&F(6#F*F*&F(6#%\"nGF2F*F2" }{TEXT -1 10 " + ... + " }{XPPEDIT
18 0 "f(z[n])/((z[n]-z[0])*`...`*(z[n]-z[n-1]));" "6#*&-%\"fG6#&%\"zG6
#%\"nG\"\"\"*(,&&F(6#F*F+&F(6#\"\"!!\"\"F+%$...GF+,&&F(6#F*F+&F(6#,&F*
F+F+F3F3F+F3" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 11 "Note that \n" }}{PARA 15 "" 0 "" {TEXT -1 42 "a
ll those numerators are algebraic numbers" }}{PARA 15 "" 0 "" {TEXT
-1 44 "all those denominators are Gaussian integers" }}{PARA 15 "" 0 "
" {TEXT -1 50 "thus every term in that sum is an algebraic number" }}
{PARA 15 "" 0 "" {TEXT -1 5 "thus " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#
%\"nG" }{TEXT -1 23 " is an algebraic number" }}{PARA 262 "" 0 ""
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "By choosing the circle \+
" }{XPPEDIT 18 0 "C[n];" "6#&%\"CG6#%\"nG" }{TEXT -1 7 " to be " }
{XPPEDIT 18 0 "abs(z) = n;" "6#/-%$absG6#%\"zG%\"nG" }{TEXT -1 1 " " }
}{PARA 15 "" 0 "" {TEXT -1 22 "it can be argued that " }{XPPEDIT 18 0
"A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 12 " is at most " }{XPPEDIT 18 0 "
e^(-n*log(n)+O(n));" "6#)%\"eG,&*&%\"nG\"\"\"-%$logG6#F'F(!\"\"-%\"OG6
#F'F(" }{TEXT -1 11 ", where \"O(" }{TEXT 707 1 "n" }{TEXT -1 23 ")\" \+
means a function of " }{TEXT 708 1 "n" }{TEXT -1 40 " whose size is at
most a constant times " }{TEXT 709 1 "n" }{TEXT -1 9 ", whereas" }}
{PARA 15 "" 0 "" {TEXT -1 48 "it can also be argued (and here I have t
o cut a " }{TEXT 702 17 "very large corner" }{TEXT -1 21 ") that that \+
whenever " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 4 " is \+
" }{TEXT 703 8 "not zero" }{TEXT -1 21 " it must be at least " }
{XPPEDIT 18 0 "e^(-n*log(n)/2+O(n));" "6#)%\"eG,&*(%\"nG\"\"\"-%$logG6
#F'F(\"\"#!\"\"F--%\"OG6#F'F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 60 "Those minimum and maximum values are inco
mpatible for large " }{TEXT 710 1 "n" }{TEXT -1 22 ", and it follows t
hat " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 4 " is " }
{TEXT 704 10 "eventually" }{TEXT -1 11 " zero. Thus" }{XPPEDIT 18 0 "e
^(Pi*z);" "6#)%\"eG*&%#PiG\"\"\"%\"zGF'" }{TEXT -1 21 " is a polynomia
l, of " }{TEXT 706 13 "finite degree" }{TEXT -1 36 " (as a consequence
of assuming that " }{XPPEDIT 18 0 "e^Pi;" "6#)%\"eG%#PiG" }{TEXT -1
20 " is algebraic). But " }{XPPEDIT 18 0 "e^(Pi*z)" "6#)%\"eG*&%#PiG\"
\"\"%\"zGF'" }{TEXT -1 20 " is most definitely " }{TEXT 705 3 "not" }
{TEXT -1 37 " a polynomial of finite degree. Thus " }{XPPEDIT 18 0 "e^
Pi" "6#)%\"eG%#PiG" }{TEXT -1 25 " is transcendental. [end]" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 351 "Suffice it to \+
say that complete proofs (Gelfond's or Schneider's) of the full versio
n of Hilbert's seventh problem are really quite difficult, and have no
place here. I will only remark that both proofs make use - as does th
e one above - of functions of a complex variable. An interested reader
will find a very good exposition in Niven's MAA classic " }{TEXT 740
18 "Irrational Numbers" }{TEXT -1 63 ". Also a reader might ask: is it
possible to give a proof that " }{TEXT 741 6 "avoids" }{TEXT -1 44 " \+
the use of complex variable methods in the " }{TEXT 742 4 "real" }
{TEXT -1 55 " case of Hilbert's seventh problem. In other words, if "
}{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "
0 < alpha;" "6#2\"\"!%&alphaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "alpha \+
<> 1;" "6#0%&alphaG\"\"\"" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "beta;
" "6#%%betaG" }{TEXT -1 29 " are real algebraic numbers, " }{XPPEDIT
18 0 "beta;" "6#%%betaG" }{TEXT -1 17 " irrational, can " }{XPPEDIT
18 0 "alpha^beta;" "6#)%&alphaG%%betaG" }{TEXT -1 9 " (namely " }
{XPPEDIT 18 0 "e^(beta*log[e](alpha));" "6#)%\"eG*&%%betaG\"\"\"-&%$lo
gG6#F$6#%&alphaGF'" }{TEXT -1 52 ") be proved to be transcendental by \+
using arguments " }{TEXT 743 3 "not" }{TEXT -1 283 " involving complex
variable methods? Gelfond himself gave such a proof, which formed an \+
entire chapter in the great Gelfond and Linnik classic, as I've alread
y mentioned in an earlier section. Incidentally Gelfond also gave ther
e an entirely elementary proof of the transcendence of " }{XPPEDIT 18
0 "e^alpha;" "6#)%\"eG%&alphaG" }{TEXT -1 29 " for non-zero real algeb
raic " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 "." }}}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 18 "Alan Baker (1939-)" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "What follows is in " }
{TEXT 1213 17 "no sense complete" }{TEXT -1 43 ", in fact it barely sc
ratches the surface. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 556 "Alan Baker was awarded a Fields Medal at the 1970 Int
ernational Congress of Mathematicians held in Nice, France. One day he
will surely be awarded the Abel Prize (an annual award, introduced in
2003 by the The Norwegian Academy of Science and Letters to honour No
rway's Niels Henrik Abel (1802-1829); J-P. Serre was the first recipie
nt in 2003, and M. Atiyah & I. Singer are the 2004 recipients. See www
.abelprisen.no/en/). He is a towering figure who transformed the subje
ct of transcendence, and related areas of Diophantine equations and ap
proximations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 43 "In the final paragraph of his classic book " }{TEXT 950
36 "Transcendental and Algebraic Numbers" }{TEXT -1 16 " Gelfond wrote
: " }{TEXT 951 382 "' Non-trivial lower bounds for linear forms, with \+
integral coefficients, of an arbitrary number of logarithms of algebra
ic numbers, obtained effectively by methods of the theory of transcend
ental numbers, will be of extraordinarily great significance in the so
lution of very difficult problems of modern number theory.Therefore, o
ne may assume, as was already mentioned above, that " }{TEXT 1214 68 "
the most pressing problem in the theory of transcendental numbers is"
}{TEXT 1215 104 " the investigation of the measures of transcendence o
f finite sets of laogarithms of algebraic numbers.'" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "From Paul
Tur\341n's talk" }{TEXT 1183 1 " " }{TEXT -1 79 "1970 at the 1970 Int
ernational Congress of Mathematicians I selectively quote: " }{TEXT
1107 833 "The theory of transcendental numbers, initiated by Liouville
in 1844, has been enriched greatly in recent years. Among the relevan
t profound contributions are those of A. Baker, W. M. Schmidt, and V. \+
A. Sprindzuk. Their work moves in important directions which contrast \+
with the traditional concentration on the deep problem of finding sign
ificant classes of finding functions assuming transcendental values fo
r all non-zero algebraic values of the independent variable. Among the
se, Baker's have had the heaviest impact on other problems in mathemat
ics. Perhaps the most significant of these impacts has been the applic
ation to diophantine equations. This theory, carrying a history of mor
e than a thousand years, was, until the early years of this century, l
ittle more than a collection of isolated problems subjected to ingenio
us " }{TEXT -1 6 "ad hoc" }{TEXT 1186 10 " methods. " }{TEXT -1 0 "" }
{TEXT 1184 582 "It was A. Thue who made the breakthrough to general re
sults by proving in 1909 that all diophantine equations of the form f(
x, y) = m, where m is an integer and f is an irreducible homogeneous f
orm of degree at least three, with integer coefficients, has at most a
finite number of solutions in integers. This theorem was extended by \+
C. L. Siegel and K. F. Roth (himself a Fields medallist) to much more \+
general classes of algebraic diophantine equations in two variables of
degree at least three. They even succeeded in establishing upper boun
ds on the number of such solutions. A " }{TEXT -1 8 "complete" }{TEXT
1187 63 " resolution of such problems however, requiring a knowledge o
f " }{TEXT -1 3 "all" }{TEXT 1188 101 " solutions, is basically beyond
the reach of these methods, which are what are called \"ineffective\"
. " }{TEXT 1185 35 "Here Baker made a brilliant advance" }{TEXT 1189
6 ". ... " }{TEXT -1 89 "[Tur\341n continues to list one development a
fter another, and links to the work of others] " }{TEXT 1190 30 "As an
other consequence of his " }{TEXT -1 9 "[Baker's]" }{TEXT 1192 20 " re
sults he gave an " }{TEXT 1191 9 "effective" }{TEXT 1193 120 " lower b
ound for the approximability of algebraic numbers by rationals, the fi
rst one which is better than Liouville's. " }}{PARA 0 "" 0 "" {TEXT
1194 178 " As mentioned before, these results are all consequences \+
of his main results on transcendental numbers. As is well known, the s
eventh problem of Hilbert asking whether or not " }{XPPEDIT 18 0 "alph
a^beta;" "6#)%&alphaG%%betaG" }{TEXT 1196 28 " is transcendental whene
ver " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 1 " " }{TEXT 1197 171 "are
algebraic, certain obvious cases aside, was solved independently by G
elfond and Schneider in 1934. Shortly afterwards Gelfond found a stron
ger result by obtaining and " }{TEXT -1 8 "explicit" }{TEXT 1199 17 " \+
lower bound for " }{XPPEDIT 18 0 "abs(beta[1]*log(alpha[1])+beta[2]*lo
g(alpha[2]));" "6#-%$absG6#,&*&&%%betaG6#\"\"\"F+-%$logG6#&%&alphaG6#F
+F+F+*&&F)6#\"\"#F+-F-6#&F06#F5F+F+" }{TEXT -1 1 " " }{TEXT 1198 173 "
in terms of the alphas and the degrees and heights of the betas when t
he logarithms are linearly independent (JC comment: over the rationals
, and thus over the algebraics). " }{TEXT -1 0 "" }{TEXT 1195 95 "Afte
r Gelfond realised in 1948, in collaboration with Ju. V. Linnik, the s
ignificance of an an " }{TEXT -1 9 "effective" }{TEXT 1201 116 " lower
bound for the three-term sum, and more generally for the n-term sum, \+
he and N. I. Feldman soon discovered an " }{TEXT -1 11 "ineffective" }
{TEXT 1202 70 " lower bound for it. The transition from this important
first step to " }{TEXT -1 9 "effective" }{TEXT 1203 238 " lower bound
for the three-term sum, and more generally for the n-term sum, resist
ed all efforts until Baker's success in 1966. This success enabled Bak
er to obtain a vast generalization of Gelfond-Schneider's theorem by s
howing that if " }{XPPEDIT 18 0 "alpha[1],alpha[2],`...`,alpha[n];" "
6&&%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 1 " " }{TEXT
1200 28 "(none 0 or 1) are algebraic," }{TEXT -1 1 " " }{XPPEDIT 18 0
"beta[1],beta[2],`...`,beta[n];" "6&&%%betaG6#\"\"\"&F$6#\"\"#%$...G&F
$6#%\"nG" }{TEXT -1 1 " " }{TEXT 1204 3 "are" }{TEXT -1 1 " " }{TEXT
1205 14 "algebraic with" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1,beta[1],bet
a[2],`...`,beta[n];" "6'\"\"\"&%%betaG6#F#&F%6#\"\"#%$...G&F%6#%\"nG"
}{TEXT -1 1 " " }{TEXT 1206 45 "linearly independent over the rational
s, then" }{TEXT -1 1 " " }{XPPEDIT 18 0 "alpha[1]^beta[1]*alpha[2]^bet
a[2]*`...`*alpha[n]^beta[n];" "6#**)&%&alphaG6#\"\"\"&%%betaG6#F(F()&F
&6#\"\"#&F*6#F/F(%$...GF()&F&6#%\"nG&F*6#F6F(" }{TEXT -1 1 " " }{TEXT
1207 52 "is transcendental. Some further appreciation of the " }{TEXT
1208 5 "depth" }{TEXT 1209 157 " of this result can be gained by recal
ling Hilbert's prediction that the Riemann hypothesis conjecture would
be settled long before the transcendentality of " }{XPPEDIT 18 0 "alp
ha^beta;" "6#)%&alphaG%%betaG" }{TEXT -1 2 ". " }{TEXT 1210 205 "The a
nalytic prowess displayed by Baker could hardly receive a higher testi
monial... Among his other results generalizing transcendentality resul
ts of Siegel and Schneider I mention only one special case, " }{TEXT
1211 33 "in itself sufficiently remarkable" }{TEXT 1212 120 ", accordi
ng to which the sum of circumferences of two ellipses, whose axes have
algebraic lengths, is transcendental... " }}{PARA 258 "" 0 "" {TEXT
-1 18 "__________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 54 "In the early 1960s, before Baker's stunning series
of " }{TEXT 967 51 "linear forms in the logarithms of algebraic numbe
rs" }{TEXT -1 146 " papers began to appear, Baker (while a PhD student
at Cambridge) proved many remarkable results, of which I choose just \+
one, and its consequence:" }}{PARA 0 "" 0 "" {TEXT -1 49 "Baker's Theo
rem (1964). For all rational numbers " }{XPPEDIT 18 0 "p/q;" "6#*&%\"p
G\"\"\"%\"qG!\"\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "0 < q;" "6#2\"\"!%
\"qG" }{TEXT -1 10 ") we have\n" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "abs(2^(1/3)-p/q);" "6#-%$absG6#,&)\"\"#*&\"\"\"F*\"\"$!
\"\"F**&%\"pGF*%\"qGF,F," }{TEXT -1 1 " " }{TEXT 955 1 ">" }{TEXT -1
1 " " }{XPPEDIT 18 0 "10^(-6)/(q^2.995);" "6#*&)\"#5,$\"\"'!\"\"\"\"\"
)%\"qG$\"%&*H!\"$F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 93 "Baker's immediate consequence. Let N be any non-zero in
teger, then all solutions in integers " }{TEXT 956 1 "x" }{TEXT -1 2 "
, " }{TEXT 957 1 "y" }{TEXT -1 17 " of the equation " }}{PARA 258 ""
0 "" {XPPEDIT 18 0 "x^3-2*y^3 = N;" "6#/,&*$%\"xG\"\"$\"\"\"*&\"\"#F(*
$%\"yGF'F(!\"\"%\"NG" }{TEXT -1 8 " ... (3)" }}{PARA 0 "" 0 "" {TEXT
-1 13 "satisfy max(|" }{TEXT 959 1 "x" }{TEXT -1 4 "|, |" }{TEXT 960
1 "y" }{TEXT -1 3 "|) " }{TEXT 958 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18
0 "(300000*abs(N))^23;" "6#*$*&\"'++I\"\"\"-%$absG6#%\"NGF&\"#B" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 123 "Appreciation. To feel the value of that result one shoul
d recall Thue's theorem would inform only that there are at most a " }
{TEXT 966 6 "finite" }{TEXT -1 20 " number of integers " }{TEXT 961 1
"x" }{TEXT -1 2 ", " }{TEXT 962 2 "y " }{TEXT -1 99 "satisfying (3). A
lso one should remind oneself of the fact that in the quadratic case o
f (3) (i.e. " }{XPPEDIT 18 0 "x^2-2*y^2 = N;" "6#/,&*$%\"xG\"\"#\"\"\"
*&F'F(*$%\"yGF'F(!\"\"%\"NG" }{TEXT -1 55 " ... (2)) matters are quite
different: for any integer " }{TEXT 963 2 "N " }{TEXT -1 28 "the equa
tion (2) either had " }{TEXT 964 2 "no" }{TEXT -1 55 " solutions (e.g.
, there are no integers x, y such that " }{XPPEDIT 18 0 "x^2-2*y^2 = 3
;" "6#/,&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(!\"\"\"\"$" }{TEXT -1 31 "
; that's a simple exercise) or " }{TEXT 965 15 "infinitely many" }
{TEXT -1 18 " solutions (e.g., " }{XPPEDIT 18 0 "x^2-2*y^2 = 1;" "6#/,
&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(!\"\"F(" }{TEXT -1 32 " has infini
tely many solutions)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 57 "Now, a question related to the Gelfond-Schneider theor
em." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Th
eorem (Baker, 1966). If " }{XPPEDIT 18 0 "alpha[1],alpha[2],`...`,alph
a[n];" "6&&%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 42 " \+
are non-zero algebraic numbers such that " }{XPPEDIT 18 0 "log(alpha[1
]),log(alpha[2]),`...`,log(alpha[n]);" "6&-%$logG6#&%&alphaG6#\"\"\"-F
$6#&F'6#\"\"#%$...G-F$6#&F'6#%\"nG" }{TEXT -1 135 " [no matter what va
lues are chosen of the logarithms are chosen in the complex case] are \+
linearly independent over the rationals, then " }{XPPEDIT 18 0 "1,log(
alpha[1]),log(alpha[2]),`...`,log(alpha[n]);" "6'\"\"\"-%$logG6#&%&alp
haG6#F#-F%6#&F(6#\"\"#%$...G-F%6#&F(6#%\"nG" }{TEXT -1 66 " are linear
ly independent over the field of all algebraic numbers." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 1170 12 "Consequences" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 166 "Theorem (Baker, 1966). Any non-vanishing linear combinat
ion of logarithms of algebraic numbers with algebraic coefficients is \+
transcendental. \n In other words, if " }{XPPEDIT 18 0 "alpha[1],al
pha[2],`...`,alpha[n];" "6&&%&alphaG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG
" }{TEXT -1 37 " are non-zero algebraic numbers, and " }{XPPEDIT 18 0
"beta[1],beta[2],`...`,beta[n];" "6&&%%betaG6#\"\"\"&F$6#\"\"#%$...G&F
$6#%\"nG" }{TEXT -1 33 " are algebraic numbers such that " }{XPPEDIT
18 0 "beta[1]*log(alpha[1])+beta[2]*log(alpha[2])+`...`+beta[n]*log(al
pha[n]) <> 0;" "6#0,**&&%%betaG6#\"\"\"F)-%$logG6#&%&alphaG6#F)F)F)*&&
F'6#\"\"#F)-F+6#&F.6#F3F)F)%$...GF)*&&F'6#%\"nGF)-F+6#&F.6#F 0,1;" "6$0&%&alphaG6#\"\"\"\"\"!F'" }{TEXT -1 6 ", and " }
{XPPEDIT 18 0 "beta[1];" "6#&%%betaG6#\"\"\"" }{TEXT -1 48 " is irrati
onal. Now consider two such numbers - " }{XPPEDIT 18 0 "alpha[1]^beta[
1]" "6#)&%&alphaG6#\"\"\"&%%betaG6#F'" }{TEXT -1 5 " and " }{XPPEDIT
18 0 "alpha[2]^beta[2];" "6#)&%&alphaG6#\"\"#&%%betaG6#F'" }{TEXT -1
98 " - and consider their product and sum. What can be said about tho
se numbers? (Bear in mind it is " }{TEXT 1309 21 "completely elementar
y" }{TEXT -1 9 " that if " }{XPPEDIT 18 0 "t[1],t[2];" "6$&%\"tG6#\"\"
\"&F$6#\"\"#" }{TEXT -1 37 " are two transcendental numbers then " }
{TEXT 1308 12 "at least one" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "t[1]+t
[2];" "6#,&&%\"tG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "t[1]*t[2];" "6#*&&%\"tG6#\"\"\"F'&F%6#\"\"#F'" }{TEXT
-1 54 " must be transcendental; recall the remark concerning " }
{XPPEDIT 18 0 "e+Pi;" "6#,&%\"eG\"\"\"%#PiGF%" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "e*Pi;" "6#*&%\"eG\"\"\"%#PiGF%" }{TEXT -1 361 " at the \+
end of the Siegel section). In reality, nothing of any value can be sa
id about their sum, and I won't dwell on it, but something profound ca
n be said about their product. Essentially what I am going to say is t
hat the product is transcendental, but one needs to rule out of consid
eration one obvious case, which I hope is made clear by these two case
s: " }}{PARA 15 "" 0 "" {TEXT -1 7 "Choose " }{XPPEDIT 18 0 "alpha[1],
beta[1],alpha[2],beta[2];" "6&&%&alphaG6#\"\"\"&%%betaG6#F&&F$6#\"\"#&
F(6#F," }{TEXT -1 7 " to be " }{XPPEDIT 18 0 "5,sqrt(3),25,-sqrt(3)/2;
" "6&\"\"&-%%sqrtG6#\"\"$\"#D,$*&-F%6#F'\"\"\"\"\"#!\"\"F/" }{TEXT -1
7 ", then " }{XPPEDIT 18 0 "alpha[1]^beta[1]" "6#)&%&alphaG6#\"\"\"&%%
betaG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha[2]^beta[2];" "6#)
&%&alphaG6#\"\"#&%%betaG6#F'" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "5^sq
rt(3);" "6#)\"\"&-%%sqrtG6#\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
25^(-sqrt(3)/2);" "6#)\"#D,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F-" }
{TEXT -1 20 ",\nand their product " }{XPPEDIT 18 0 "5^sqrt(3)" "6#)\"
\"&-%%sqrtG6#\"\"$" }{TEXT -1 1 "." }{XPPEDIT 18 0 "25^(-sqrt(3)/2)" "
6#)\"#D,$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"F-" }{TEXT -1 166 "is clearl
y 1, is most certainly not transcendental!\nOf course you see how I ma
de that happen: I choose the 5 and 25 in the obvious way, and the two \+
irrational powers " }{XPPEDIT 18 0 "sqrt(3),-sqrt(3)/2;" "6$-%%sqrtG6#
\"\"$,$*&-F$6#F&\"\"\"\"\"#!\"\"F-" }{TEXT -1 70 " are linearly depend
ent over the rationals in such a way as to make..." }}{PARA 15 "" 0 "
" {TEXT -1 86 "Here I make a choice that is like the previous one, but
with one small change: choose " }{XPPEDIT 18 0 "alpha[1],beta[1],alph
a[2],beta[2];" "6&&%&alphaG6#\"\"\"&%%betaG6#F&&F$6#\"\"#&F(6#F," }
{TEXT -1 7 " to be " }{XPPEDIT 18 0 "5,sqrt(3),25,1/3-sqrt(3)/2;" "6&
\"\"&-%%sqrtG6#\"\"$\"#D,&*&\"\"\"F+F'!\"\"F+*&-F%6#F'F+\"\"#F,F," }
{TEXT -1 7 ", then " }{XPPEDIT 18 0 "alpha[1]^beta[1]" "6#)&%&alphaG6#
\"\"\"&%%betaG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "alpha[2]^beta[
2];" "6#)&%&alphaG6#\"\"#&%%betaG6#F'" }{TEXT -1 5 " are " }{XPPEDIT
18 0 "5^sqrt(3);" "6#)\"\"&-%%sqrtG6#\"\"$" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "25^(1/3-sqrt(3)/2);" "6#)\"#D,&*&\"\"\"F'\"\"$!\"\"F'*&
-%%sqrtG6#F(F'\"\"#F)F)" }{TEXT -1 18 "and their product " }{XPPEDIT
18 0 "5^sqrt(3)" "6#)\"\"&-%%sqrtG6#\"\"$" }{TEXT -1 1 "." }{XPPEDIT
18 0 "25^(1/3-sqrt(3)/2);" "6#)\"#D,&*&\"\"\"F'\"\"$!\"\"F'*&-%%sqrtG6
#F(F'\"\"#F)F)" }{TEXT -1 11 "is clearly " }{XPPEDIT 18 0 "25^(1/3);"
"6#)\"#D*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 105 " , is clearly algebraic.
\nOf course you again see how I made that happen: this time the irrati
onal powers " }{XPPEDIT 18 0 "sqrt(3),1/3-sqrt(3)/2;" "6$-%%sqrtG6#\"
\"$,&*&\"\"\"F)F&!\"\"F)*&-F$6#F&F)\"\"#F*F*" }{TEXT -1 1 " " }{TEXT
1311 3 "are" }{TEXT -1 11 " linearly \n" }{TEXT 1310 2 "in" }{TEXT -1
52 "dependent over the rationals, but the three numbers " }{TEXT 1312
1 "1" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sqrt(3), 1/3-sqrt(3)/2" "6$-%%s
qrtG6#\"\"$,&*&\"\"\"F)F&!\"\"F)*&-F$6#F&F)\"\"#F*F*" }{TEXT -1 43 " a
re linearly dependent over the rationals." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }{TEXT 1124 189 "To end, I quot
e directly from two papers of Alan Baker (the celebrated 1968 pair, pu
blished back-to-back in the same issue of the Philosophical Transactio
ns of the Royal Society of London):" }{TEXT -1 0 "" }}{PARA 258 "" 0 "
" {TEXT -1 117 "CONTRIBUTIONS TO THE THEORY OF DIOPHANTINE EQUATIONS\n
1. ON THE REPRESENTATION OF INTEGERS BY BINARY FORMS\nBy A. BAKER" }}
{PARA 258 "" 0 "" {TEXT -1 15 "1. Introduction" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "It was proved by Thue (19
09) that the Diophantine equation" }}{PARA 258 "" 0 "" {TEXT -1 1 " "
}{TEXT 1109 1 "f" }{TEXT -1 1 "(" }{TEXT 1110 1 "x" }{TEXT -1 2 ", " }
{TEXT 1111 1 "y" }{TEXT -1 4 ") = " }{TEXT 1112 1 "m" }{TEXT -1 15 ", \+
(1)" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 1113 1 "f
" }{TEXT -1 89 " denotes an irreducible binary form with integer coeff
icients and degree at least 3, and " }{TEXT 1120 1 "m" }{TEXT -1 73 " \+
is any integer, possesses only a finite number of solutions in integer
s " }{TEXT 1115 1 "x" }{TEXT -1 2 ", " }{TEXT 1114 1 "y" }{TEXT -1
443 ". Thue discovered the theorem by way of his fundamental studies o
n rational approximations to algebraic numbers, which were later profo
undly developed in the celebrated works of Siegel (1921) and Roth (195
5), and which formed the genesis of many other investigations. But Thu
e's theorem, like all subsequent developments, suffers from one basic \+
limitation, that of its non-effectiveness... Indeed it would seem that
even for cubic polynomials " }{TEXT 1121 1 "f" }{TEXT -1 212 ", no ge
nerally effective algorithm for the complete solution of (1) has hithe
rto been established, although a wide variety of techniques have been \+
successfully employed to treat particular equations of this kind." }}
{PARA 0 "" 0 "" {TEXT -1 197 " The present paper is devoted to a ne
w proof of Thue's theorem, which proceeds by an argument that is effec
tive, and therefore provides an algorithm for the complete solution of
(1) in integers " }{TEXT 1122 1 "x" }{TEXT -1 2 ", " }{TEXT 1123 1 "y
" }{TEXT -1 6 ". Let " }{TEXT 1127 1 "f" }{TEXT -1 1 "(" }{TEXT 1128
1 "x" }{TEXT -1 2 ", " }{TEXT 1129 1 "y" }{TEXT -1 37 ") denote a homo
geneous polynomial in " }{TEXT 1130 1 "x" }{TEXT -1 2 ", " }{TEXT
1131 1 "y" }{TEXT -1 13 " with degree " }{TEXT 1125 1 "n" }{TEXT -1
89 " (at least 3) and with integer coefficients, irreducible over the \+
rationals. Suppose that" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "kappa;" "6
#%&kappaG" }{TEXT -1 3 " > " }{TEXT 1126 1 "n" }{TEXT -1 17 " + 1 \+
(2)" }}{PARA 0 "" 0 "" {TEXT -1 8 "and let " }{TEXT 1132 1 "m" }
{TEXT -1 69 " be any positive integer. The main result of this paper i
s as follows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 11 "Theorem 1. " }{TEXT 1133 17 "All solutions of " }{TEXT -1 3 "(1
)" }{TEXT 1134 25 " in integers x, y satisfy" }}{PARA 258 "" 0 ""
{TEXT -1 5 "max(|" }{TEXT 1135 1 "x" }{TEXT -1 4 "|, |" }{TEXT 1136 1
"y" }{TEXT -1 5 "|) < " }{XPPEDIT 18 0 "C*e^(log(m)^kappa);" "6#*&%\"C
G\"\"\")%\"eG)-%$logG6#%\"mG%&kappaGF%" }{TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 6 "where " }{TEXT 1137 1 "C" }{TEXT -1 56 " is an effectiv
ely computabe constant depending only on " }{TEXT 1138 1 "n" }{TEXT
-1 2 ", " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 26 ", and the
coefficients of " }{TEXT 1139 1 "f" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 11 "... ... ..." }}{PARA 0 "" 0 "" {TEXT -1 11 "Theorem 2. \+
" }{TEXT 1140 8 "Suppose " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }
{TEXT -1 1 " " }{TEXT 1141 60 "is an algebraic number with degree n (a
t least 3), and that " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1
3 " > " }{TEXT 1142 1 "n" }{TEXT -1 6 " + 1. " }{TEXT 1145 16 "Then th
ere is an" }{TEXT -1 1 " " }{TEXT 1146 29 "effectively computable numb
er" }{TEXT -1 1 " " }{TEXT 1143 2 "c " }{TEXT -1 3 " = " }{XPPEDIT 18
0 "c(alpha,kappa);" "6#-%\"cG6$%&alphaG%&kappaG" }{TEXT -1 5 " > 0 " }
{TEXT 1147 9 "such that" }{TEXT -1 1 "\n" }}{PARA 258 "" 0 ""
{XPPEDIT 18 0 "abs(alpha-p/q);" "6#-%$absG6#,&%&alphaG\"\"\"*&%\"pGF(%
\"qG!\"\"F," }{TEXT -1 1 " " }{TEXT 1144 1 ">" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c*q^(-n)*e^(log(q)^(1/kappa));" "6#*(%\"cG\"\"\")%\"qG,
$%\"nG!\"\"F%)%\"eG)-%$logG6#F'*&F%F%%&kappaGF*F%" }{TEXT -1 10 " \+
(3)" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 1148 18 "for all integ
ers p" }{TEXT -1 1 "," }{TEXT 1150 3 " q " }{TEXT -1 1 "(" }{TEXT
1149 1 "q" }{TEXT -1 5 " > 0)" }{TEXT 1151 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "And later:" }}{PARA 258 "
" 0 "" {TEXT -1 82 "CONTRIBUTIONS TO THE THEORY OF DIOPHANTINE EQUATIO
NS\nII. THE DIOPHANTINE EQUATION " }{XPPEDIT 18 0 "y^2 = x^3+k;" "6#/*
$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"%\"kGF+" }{TEXT -1 12 "\nBy A. BAKER" }
}{PARA 258 "" 0 "" {TEXT -1 15 "1. Introduction" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "The problem of finding t
he totality of integers whose cubes differ by a given integer from a s
quare has interested mathematicians for several centuries. According t
o Dickson's " }{TEXT 1152 32 "History of the theory of numbers" }
{TEXT -1 413 " researches on the subject can be traced back at least a
s far as Bachet (1621), and numerous contributions to the general theo
ry can be found in the works of Fermat, Euler, V. A. Lebesgue, Pepin, \+
Jonqui\350res and many others. More especially, during the past fifty \+
or so years, the equation of the title has been extensively investigat
ed by Mordell (1913, 1914, 1922, 1923, 1963, 1964 and see, in particul
ar, 1947: " }{TEXT 1153 34 "A chapter in the theory of numbers" }
{TEXT -1 286 "), Nagell (1929, 1930), Delaunay (1929) (Footnote: See a
lso Delone [Delaunay] & Faddeev (1940)), Ljunggren (1942, 1963) and He
mer (1952, 1954) (Footnote: See also Skolem (1938), Marshall Hall (195
3)), and a complete set of solutions has now been obtained for large c
lasses of values of " }{TEXT 1154 1 "k" }{TEXT -1 34 "; these include,
for example, all " }{TEXT 1155 1 "k" }{TEXT -1 16 " satisfying 0 < "
}{XPPEDIT 18 0 "abs(k) <= 100;" "6#1-%$absG6#%\"kG\"$+\"" }{TEXT -1
93 ", except for 20 special cases. The methods of solution vary widely
according to the specific " }{TEXT 1156 1 "k" }{TEXT -1 295 " under d
iscussion, but they usually involve a combination of congruence techni
ques, together with a detailed study of the arithmetic of certain unde
rlying number fields. In addition the argument often utilizes the well
known connection between the equation of the title and equations of t
he kind" }}{PARA 258 "" 0 "" {TEXT 1157 5 " f" }{TEXT -1 1 "(" }
{TEXT 1158 1 "x" }{TEXT -1 2 ", " }{TEXT 1159 1 "y" }{TEXT -1 4 ") = \+
" }{TEXT 1160 1 "m" }{TEXT -1 15 ", (1)" }}{PARA 0 "" 0 ""
{TEXT -1 6 "where " }{TEXT 1161 1 "f" }{TEXT -1 89 " denotes an irredu
cible binary form with integer coefficients and degree at least 3, and
" }{TEXT 1162 1 "m" }{TEXT -1 234 " denotes a fixed integer. In parti
cular Mordell (1922, 1923) (Footnote: See also Thue (1917), Landau & O
strowski (1920), Siegel (1926)) has employed this feature, together wi
th the famous theorem of Thue (1909), to show that, for any " }
{XPPEDIT 18 0 "k <> 0;" "6#0%\"kG\"\"!" }{TEXT -1 14 ", the equation"
}}{PARA 258 "" 0 "" {XPPEDIT 18 0 "y^2 = x^3+k" "6#/*$%\"yG\"\"#,&*$%
\"xG\"\"$\"\"\"%\"kGF+" }{TEXT -1 13 " (2)" }}{PARA 0 "" 0 "
" {TEXT -1 50 "has only a finite number of solutions in integers " }
{TEXT 1163 1 "x" }{TEXT -1 2 ", " }{TEXT 1164 1 "y" }{TEXT -1 134 ". N
o general algorithm, however, has hitherto been established which woul
d enable one to find all solutions of (2) for any prescribed " }{TEXT
1165 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 316 " In a re
cent paper (Baker 1968 [JC: the first paper above]) a new proof of the
finiteness of the number of solutions of (1) was given, which, in con
trast to Thue's original proof, proceeds by an argument that is effect
ive and provides therefore a process for determining all solutions of \+
the equation in integers " }{TEXT 1166 1 "x" }{TEXT -1 2 ", " }{TEXT
1167 1 "y" }{TEXT -1 154 ". Moreover, it was remarked that, in consequ
ence of this result, one could now obtain an effective algorithm for t
he complete solution of (2) in integers " }{TEXT 1168 1 "x" }{TEXT -1
2 ", " }{TEXT 1169 1 "y" }{TEXT -1 147 ", and it is the purpose of the
present paper to supply the details of the demonstration. The precise
result that will be established is as follows." }}{PARA 0 "" 0 ""
{TEXT -1 27 "Theorem 1. For any integer " }{XPPEDIT 18 0 "k <> 0;" "6#
0%\"kG\"\"!" }{TEXT -1 15 ", and integers " }{TEXT 1116 1 "x" }{TEXT
-1 2 ", " }{TEXT 1117 2 "y " }{TEXT -1 5 "with " }{XPPEDIT 18 0 "y^2 =
x^3+k;" "6#/*$%\"yG\"\"#,&*$%\"xG\"\"$\"\"\"%\"kGF+" }{TEXT -1 15 ", \+
we have max(|" }{TEXT 1118 1 "x" }{TEXT -1 4 "|, |" }{TEXT 1119 1 "y"
}{TEXT -1 5 "|) < " }{XPPEDIT 18 0 "e^(10^10*abs(k)^(10^4));" "6#)%\"e
G*&\"#5F&)-%$absG6#%\"kG*$F&\"\"%\"\"\"" }{TEXT -1 1 "." }}{PARA 258 "
" 0 "" {TEXT -1 11 "___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 102 "I could report on later developments, bu
t I need to end somewhere, and here is as good a place as any." }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Appendix on the " }{TEXT 526 4 "a
bc-" }{TEXT -1 10 "conjecture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 105 "I know many of you attended the full-hou
se public lecture - sponsored by the Royal Irish Academy and the " }
{TEXT 1303 11 "Irish Times" }{TEXT -1 153 " - in Trinity College Dubli
n last October 14th, given by Andrew Wiles (who, as the saying goes, n
eeds no introduction). In his talk, Wiles mentioned the " }{TEXT 1304
14 "abc-conjecture" }{TEXT -1 89 ", and I thought there might be some \+
interest because of that in this small final section." }}{PARA 258 ""
0 "" {TEXT -1 16 "________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 260 "A great new conjecture has been formulat
ed in recent times (since the mid 1980s) by Oesterl\351 and Masser. Th
ere are many outstanding web references (I especially recommend Abderr
ahmane Nitaj's), and there is the highly motivating Nov 2002 AMS Notic
es article - " }{TEXT 1273 19 "It's as easy as abc" }{TEXT -1 27 " - b
y Granville and Tucker." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 102 "Here, before stating the conjecture, my reader may \+
be captivated by two well-known consequences of it:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 61 "Fermat's Last Theorem \+
(so well known because of Andrew Wiles)" }}{PARA 15 "" 0 "" {TEXT -1
99 "Roth's 1955 rational approximation to algebraic numbers theorem (t
hat this is a consequence of the " }{TEXT 1272 3 "abc" }{TEXT -1 108 "
-conjecture was established in 1994 by the renowned Italian mathematic
ian, Fields medallist Enrico Bombieri)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 194 "Of course these consequences have al
ready been independently established, but there are many, many remarka
ble consequences that are still open questions (Nitaj maintains a list
of consequences)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }{TEXT 1252 4 "The " }{TEXT 1256 3 "abc" }{TEXT 1257 11
"-conjecture" }{TEXT 1254 0 "" }{TEXT 1255 0 "" }{TEXT -1 6 ". Let " }
{TEXT 1258 2 "a " }{TEXT -1 4 "and " }{TEXT 1267 1 "b" }{TEXT -1 83 " \+
be relatively prime natural numbers (i.e. they have no common prime di
visor), let " }{TEXT 1268 1 "c" }{TEXT -1 15 " be their sum: " }{TEXT
1259 1 "a" }{TEXT -1 3 " + " }{TEXT 1260 1 "b" }{TEXT -1 3 " = " }
{TEXT 1261 1 "c" }{TEXT -1 10 ", and let " }{XPPEDIT 18 0 "epsilon;" "
6#%(epsilonG" }{TEXT -1 39 " be any positive number; then there is " }
{TEXT 1285 4 "some" }{TEXT -1 10 " constant " }{TEXT 1262 1 "C" }
{TEXT -1 22 " (whose value depends " }{TEXT 1286 4 "only" }{TEXT -1 4
" on " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 11 ") such t
hat" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "c < C(epsilon)*(p[1]*p[2]*`...
`*p[r])^(1+epsilon)" "6#2%\"cG*&-%\"CG6#%(epsilonG\"\"\")**&%\"pG6#F*F
*&F.6#\"\"#F*%$...GF*&F.6#%\"rGF*,&F*F*F)F*F*" }{TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "\{p[1], p[2], `...`, p[r]\};" "6#<&&%\"pG6#\"\"\"&F%6#
\"\"#%$...G&F%6#%\"rG" }{TEXT -1 9 " are the " }{TEXT 1263 8 "distinct
" }{TEXT -1 18 " prime factors of " }{TEXT 1264 1 "a" }{TEXT -1 2 ", \+
" }{TEXT 1265 1 "b" }{TEXT -1 5 " and " }{TEXT 1266 1 "c" }{TEXT -1 1
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Gra
nville and Tucker note it has been conjectured (by whom?) that the " }
{TEXT 1275 3 "abc" }{TEXT -1 28 "-conjecture holds good with " }
{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 9 " = 1 and " }
{TEXT 1274 1 "C" }{TEXT -1 23 " = 1, which would give " }{XPPEDIT 18
0 "c < (p[1]*p[2]*`...`*p[r])^2;" "6#2%\"cG*$**&%\"pG6#\"\"\"F*&F(6#\"
\"#F*%$...GF*&F(6#%\"rGF*F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT
-1 31 " By way of illustrating the " }{TEXT 1279 5 "power" }{TEXT
-1 77 " of the latter form of the conjecture I will show that the only
solutions of " }{XPPEDIT 18 0 "2^x+3^y = 5^z;" "6#/,&)\"\"#%\"xG\"\"
\")\"\"$%\"yGF()\"\"&%\"zG" }{TEXT -1 20 " in natural numbers " }
{TEXT 1276 1 "x" }{TEXT -1 2 ", " }{TEXT 1277 1 "y" }{TEXT -1 5 " and \+
" }{TEXT 1278 1 "z" }{TEXT -1 35 " would be (1, 1, 1) and (4, 2, 2). \+
" }}{PARA 0 "" 0 "" {TEXT -1 70 " First, though, my reader may wish
to see some factored values of (" }{XPPEDIT 18 0 "2^x+3^y;" "6#,&)\"
\"#%\"xG\"\"\")\"\"$%\"yGF'" }{TEXT -1 8 "), with " }{TEXT 1280 1 "x"
}{TEXT -1 5 " and " }{TEXT 1281 1 "y" }{TEXT -1 48 " varying over a sm
all range of values. Note the " }{TEXT 1284 9 "near miss" }{TEXT -1 4
" at " }{TEXT 1282 1 "x" }{TEXT -1 6 " = 8, " }{TEXT 1283 1 "y" }
{TEXT -1 27 " = 6, where ones sees that " }{XPPEDIT 18 0 "2^6+3^8 = 66
25;" "6#/,&*$\"\"#\"\"'\"\"\"*$\"\"$\"\")F(\"%Dm" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "5^3;" "6#*$\"\"&\"\"$" }{TEXT -1 5 "*53. " }}{PARA 0 "
" 0 "" {TEXT -1 30 "Other similar near misses are " }{XPPEDIT 18 0 "2^
13+3^9 = 5^3;" "6#/,&*$\"\"#\"#8\"\"\"*$\"\"$\"\"*F(*$\"\"&F*" }{TEXT
-1 7 "*223, " }{XPPEDIT 18 0 "2^20+3^10 = 5^3;" "6#/,&*$\"\"#\"#?\"\"
\"*$\"\"$\"#5F(*$\"\"&F*" }{TEXT -1 7 "*8861, " }{XPPEDIT 18 0 "2^27+3
^11 = 5^3;" "6#/,&*$\"\"#\"#F\"\"\"*$\"\"$\"#6F(*$\"\"&F*" }{TEXT -1
10 "*1075159, " }{XPPEDIT 18 0 "2^64+3^2 = 5^3;" "6#/,&*$\"\"#\"#k\"\"
\"*$\"\"$F&F(*$\"\"&F*" }{TEXT -1 22 "*(an 18-digit prime), " }
{XPPEDIT 18 0 "2^71+3^3 = 5^4;" "6#/,&*$\"\"#\"#r\"\"\"*$\"\"$F*F(*$\"
\"&\"\"%" }{TEXT -1 20 "*(a 19-digit prime)." }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 81 "for x from 6 to 8 do \nfor y from 6 to 8 do\nprint
(x, y, ifactor(2^x + 3^y))\nod od:" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 13 "Applying the " }{TEXT 1301 11 "conjectured" }{TEXT -1 1 "
" }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 6 " = 1, " }
{TEXT 1287 1 "C" }{TEXT -1 21 " = 1 form of the the " }{TEXT 1288 3 "a
bc" }{TEXT -1 41 "-conjecture to hypothetical solutions of " }
{XPPEDIT 18 0 "2^x+3^y = 5^z;" "6#/,&)\"\"#%\"xG\"\"\")\"\"$%\"yGF()\"
\"&%\"zG" }{TEXT -1 12 " would give " }{XPPEDIT 18 0 "5^z < 30^2;" "6#
2)\"\"&%\"zG*$\"#I\"\"#" }{TEXT -1 28 " ('30' being 2*3*5), and so " }
{TEXT 1289 1 "z" }{TEXT -1 33 " could only be 1, 2, or 3. Then (" }
{TEXT 1293 1 "x" }{TEXT -1 2 ", " }{TEXT 1294 1 "y" }{TEXT -1 2 ", " }
{TEXT 1295 1 "z" }{TEXT -1 89 ") = (1, 1, 1) and (4, 2, 2) would be th
e only solutions. (You probably want to ask me...)" }}{PARA 258 "" 0 "
" {TEXT -1 19 "__________________\n" }}{PARA 0 "" 0 "" {TEXT -1 64 "Fi
nally, my reader may be interested in a theorem of mine which " }
{TEXT 1269 5 "looks" }{TEXT -1 14 " as though it " }{TEXT 1270 5 "ough
t" }{TEXT -1 28 " to be a consequence of the " }{TEXT 1271 3 "abc" }
{TEXT -1 17 "-conjecture, but " }{TEXT 1292 5 "isn't" }{TEXT -1 74 " (
I leave it as an exercise to see that my theorem isn't a consequence o
f " }{TEXT 1302 3 "abc" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 126 "Theorem (motivated by reflecting on
Euclid's proof of the infinitude of primes, MAA Monthly, Vol. 96, No.
4, April 1989). Let " }{XPPEDIT 18 0 "p[1],p[2],`...`,p[n];" "6&&%\"p
G6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 14 " be the first " }
{TEXT 1290 1 "n" }{TEXT -1 20 " prime numbers, let " }{XPPEDIT 18 0 "p
[i];" "6#&%\"pG6#%\"iG" }{TEXT -1 37 " be any one of those primes, and
let " }{XPPEDIT 18 0 "N[n,i];" "6#&%\"NG6$%\"nG%\"iG" }{TEXT -1 25 " \+
be the product of those " }{TEXT 1291 1 "n" }{TEXT -1 14 " primes, wit
h " }{XPPEDIT 18 0 "p[i];" "6#&%\"pG6#%\"iG" }{TEXT -1 14 " omitted; t
hen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18
0 "N[n,i]+1 = p[i]^m;" "6#/,&&%\"NG6$%\"nG%\"iG\"\"\"F*F*)&%\"pG6#F)%
\"mG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 18 "is impossible for " }{XPPEDIT 18 0 "4 <= n;" "6#1\"\"%%
\"nG" }{TEXT -1 13 ". (Note, for " }{TEXT 1297 1 "n" }{TEXT -1 9 " = 3
and " }{TEXT 1299 1 "i" }{TEXT -1 11 " = 1, that " }{XPPEDIT 18 0 "N[
3,1]+1;" "6#,&&%\"NG6$\"\"$\"\"\"F(F(F(" }{TEXT -1 17 " = 3*5+ 1 = 16 \+
= " }{XPPEDIT 18 0 "2^4;" "6#*$\"\"#\"\"%" }{TEXT -1 3 ".) " }}{PARA
0 "" 0 "" {TEXT -1 6 "Also, " }{XPPEDIT 18 0 "N[n,i]-1 = p[i]^m;" "6#/
,&&%\"NG6$%\"nG%\"iG\"\"\"F*!\"\")&%\"pG6#F)%\"mG" }{TEXT -1 19 " is i
mpossible for " }{XPPEDIT 18 0 "4 <= n;" "6#1\"\"%%\"nG" }{TEXT -1 13
". (Note, for " }{TEXT 1298 1 "n" }{TEXT -1 9 " = 3 and " }{TEXT 1300
1 "i" }{TEXT -1 11 " = 2, that " }{XPPEDIT 18 0 "N[3,2]-1;" "6#,&&%\"N
G6$\"\"$\"\"#\"\"\"F)!\"\"" }{TEXT -1 17 " = 2*5 - 1 = 9 = " }
{XPPEDIT 18 0 "3^2;" "6#*$\"\"$\"\"#" }{TEXT -1 2 ".)" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "A numerical illustrati
on for the case " }{TEXT 1296 1 "n" }{TEXT -1 6 " = 15:" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "n := 15: N[n] := product(ithprime(
m), m=1..n):\nfor i to n do\nN[n, i] := product(ithprime(m), m=1..n)/i
thprime(i)\nod:\nfor i to n do\n[ithprime(i), ifactor(N[n, i] + 1), if
actor(N[n, i] - 1)]\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT
-1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 206 "First, a reader - especially one interested in teaching,
one interested in introducing non-trivial ideas in a motivated way - \+
may find suitable material in various corners of my web site, especial
ly in the " }{TEXT 1176 15 "Courses I Teach" }{TEXT -1 4 " or " }
{TEXT 1177 14 "Talented Youth" }{TEXT -1 10 " sections." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "I only reference boo
ks that I have in my own personal library." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Aigner, M. & Ziegler, G., Proo
fs from the BOOK (3rd edition), Springer-Verlag (2003), 94-97." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Baker, A.
, Transcendental Number Theory, Cambridge University Press, 1975" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Baker, A
. & Masser, D., Transcendence Theory: Advanced and Applications, Proce
edings of a conference held in Cambridge in 1976, Academic Press, 1977
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 245 "Bake
r, A. & W\374stholz, G., Number Theory, transcendence and Diophantine \+
geometry in the next millennium (in MATHEMATICS: FRONTIERS AND PERSPEC
TIVES, V. Arnold, M. Atiyah, P. Lax and B. Mazur (Editors), American M
athematical Society, (2000), 1-12)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 34 "Berggren, L., Borwein, J. and P., " }
{TEXT 782 2 "Pi" }{TEXT -1 31 ": A Source Book, Springer, 1997" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "Borel, A
., Chowla, S., Herz, C.S., Iwasawa, K., Serre, J-P., Seminar on Comple
x Multiplication, Springer Lecture Notes on Mathematics, Number 21, 19
66" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Br
owder, F. (Editor), Mathematical Developments Arising From Hilbert Pro
blems, American Mathematical Society, 1976" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Brown, Dan, The Da Vinci Code, \+
Corgi Books, 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 28 "Calkin, N. and H. Wilf, H. " }{TEXT 518 24 "Recounting t
he rationals" }{TEXT -1 53 ", American Mathematical Monthly, 107 (2000
), 360-363." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 156 "Cassels, J. W. S., An Introduction to Diophantine Approximatio
n (Cambridge Tracts N0. 45), Cambridge University Press, (First printe
d 1957, reprinted 1965) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 98 "Conway, J. H., & Guy, R. K., The Book of Numbers, \+
COPERNICUS (An Imprint of Springer-Verlag), 1996" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Devaney, R. L., Chaos, Fr
actals, and Dynamics (" }{TEXT 910 35 "COMPUTER EXPERIMENTS IN MATHEMA
TICS" }{TEXT -1 106 "), Addison-Wesley, (1990) (I have a signed copy f
rom the AMS Centenary Meeting, Cincinnati, January 1994.)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Fel'dman, N. I., a
nd A. B. Shidlovskii, A. B., " }{TEXT 519 19 "The development and" }
{TEXT -1 1 " " }{TEXT 783 53 "present state of the theory of transcend
ental numbers" }{TEXT -1 96 ". (London Mathematical Society Translatio
ns) Russian Mathematical Surveys (1967), Vol. 22, 1-79." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Gelfond, A. O., Alge
braic and Transcendental Numbers. Dover, 1960" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Gelfond, A. O. and Yu. V
. Linnik, Yu. V., Elementary Methods in Analytic Number Theory, George
Allen & Unwin, 1966" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 14 "Goldfeld, D., " }{TEXT 854 58 "Gauss' class number prob
lem for imaginary quadratic fields" }{TEXT -1 78 ", Bulletin American \+
Mathematical Society, July 1985, Vol. 13, Number 1, 23-37." }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Gowers, W. T., "
}{TEXT 904 29 "THE IMPORTANCE OF MATHEMATICS" }{TEXT -1 53 ", availabl
e form his Cambridge University web site at" }}{PARA 0 "" 0 "" {TEXT
-1 27 "www.dpmms.cam.ac.uk/~wtg10/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Granville, A. and Tucker, T. J., " }
{TEXT 527 19 "It's As Easy As abc" }{TEXT -1 92 ". Notices American Ma
thematical Society, November 2002 (available on-line from www.ams.org)
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Klin
e, M., Mathematical Thought from Ancient to Modern Times. Oxford Unive
rsity Press, 1972" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 70 "Lang, S., Introduction to Transcendental Numbers, Addison
-Wesley, 1966" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 89 "Le Veque, W. J., Topics in Number Theory (Vol. 2), Addiso
n-Wesley, (Second printing 1961)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 126 "Mahler, K. Lectures on Diophantine Appro
ximations (Part 1: g-adic anumbers and Roth's theorem), University of \+
Notre Dame, 1961" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 105 "Masser, D., Elliptic Functions and Transcendence, Spring
er Lecture Notes in Mathematics, Number 437, 1975" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "Mordell, L. J., Diophant
ine Equations, Academic Press, (1969). I have a signed 1st edition dat
ed 21st August 1969, the day of the " }{TEXT 781 11 "Oxford 1969" }
{TEXT -1 50 " photo. See the Oxford 1969 corner of my web site." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Nasar, S.
, " }{TEXT 524 16 "A Beautiful Mind" }{TEXT -1 23 ". faber and faber, \+
1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "N
iven, I., Irrational Numbers, " }{TEXT 525 29 "Carus Mathematical Mono
graphs" }{TEXT -1 11 ", MAA, 1967" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 123 "Reid, C., Hilbert (includes an appreciat
ion of Hilbert's mathematical work by Hermann Weyl), George Allen & Un
wim Ltd, 1970" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 23 "Schmidt, W., Review of " }{TEXT 520 41 "Selected mathemat
ical papers of Axel Thue" }{TEXT -1 92 ", Bulletin of the American Mat
hematical Society, Vol. 84, Number 5, September 1978, 919-925." }
{TEXT 521 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 91 "Siegel, C. L., Transcendental Numbers, Princeto
n University Press, 1949 (recently reissued)" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Stollarsky, K. B., Algebraic Nu
mbers and Diophantine Approximation, Marcel Dekker Inc., 1974" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "Stollars
ky, K. B. Review of books by Baker, Mahler, and Waldschmidt, Bulletin \+
of the American Mathematical Society, Vol. 84, Number 6, November 1978
, 1370-1378." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 95 "Waldschmidt, M., Nombres transcendants, Springer Lecture Notes \+
in Mathematics, Number 402, 1975" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 204 "Whittaker and Watson. Modern Analysis. C
ambridge University Press (4th edition, 1935). (My own copy, which I b
ought in 1990, was the personal copy of the late Cornelius Lanczos. It
is stamped as follows: " }{TEXT 522 16 "PROF. C. LANCZOS" }{TEXT -1
1 " " }{TEXT 893 17 "PURDUE UNIVERSITY" }{TEXT -1 1 " " }{TEXT 523 17
"LAFAYETTE, IND., " }{TEXT -1 31 "and is signed: C. Lanczos 1938)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "W\374sth
olz, G., A Panorama of Number Theory (or The View from Baker's Garden
), Cambridge University Press, 2002" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 108 "The MacTutor History of Mathematics arc
hive at http://www-groups.dcs.st-andrews.ac.uk/~history/ is a famous \+
" }}{PARA 0 "" 0 "" {TEXT -1 185 "resource (though I have noted freque
nt errors of mathematical detail) and some quite bizzare ommisions (e.
g. there are - as yet - no entries for Wolfgang Schmidt or Theodore Sc
hneider)." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 341 0 "" }{TEXT -1 0 "
" }}}{MARK "0 0 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS
0 1 2 33 1 1 }