Conway's Look-and-Say algebraic number Perhaps the most remarkable example of an algebraic number (from a gee-whiz point of view) is provided by John Conway's Look -and- Say sequence (see the wonderful Book of Numbers by Conway & Guy): 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211121221, ... I hope you see why it's called the Look-and-Say sequence... (If you don't already know of N. J. A. Sloane's extraordinary On-Line Encyclopedia of Integer Sequences, then you simply must consult it; be prepared, however, to suffer the disappointment of thinking you've discovered a new ( interesting ) sequence: you will almost certainly find it there...). How many digits does the n -th term of that sequence have? Conway proved that it's roughly proportional to , where , is an algebraic number of degree 71, being a solution of the following irreducible degree polynomial equation (with integer coefficients): > restart; > f := x -> 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^31 - 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: > Digits := 50; # to give the value in Conway & Guy > fsolve(f(x) = 0, x = 1..2); > Digits := 10; > plot(f(x), x = (1.303)..(1.304)); >
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