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Sequences (x_k, y_k) and expressions P(x)=x+1,Q(x)=x^2+1

Source: Turkey TST 2000 P3

March 12, 2011
number theory unsolvednumber theory

Problem Statement

Let P(x)=x+1P(x)=x+1 and Q(x)=x2+1.Q(x)=x^2+1. Consider all sequences ⟨(xk,yk)⟩k∈N\langle(x_k,y_k)\rangle_{k\in\mathbb{N}} such that (x1,y1)=(1,3)(x_1,y_1)=(1,3) and (xk+1,yk+1)(x_{k+1},y_{k+1}) is either (P(xk),Q(yk))(P(x_k), Q(y_k)) or (Q(xk),P(yk))(Q(x_k),P(y_k)) for each k.k. We say that a positive integer nn is nice if xn=ynx_n=y_n holds in at least one of these sequences. Find all nice numbers.
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