p(x) is a polyonomial with rational coefficients such that k3 - k = 331992 = p(k)3 - p(k) for some real number k. Let pn(x) be p(p( ... p(x) ... )) (iterated n times). Show that pn(k)3 - pn(k) = 331992.
Solution
A rare flop. The solution is trivial.
The polynomial x3 - x has roots at 0, ±1. Obviously its values in the range -1 ≤ x ≤ 1 are between -1 and 1. Hence x3 - x = 331992 has just one real root which must be k. Hence p(k) = k. Hence pn(k) = k also.
© John Scholes
jscholes@kalva.demon.co.uk
25 Nov 2003
Last updated/corrected 25 Nov 2003