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Number of lattice point

Source: Japan Mathematical Olympiad Finals 2005, Problem 2

April 18, 2005
algebrapolynomialnumber theory proposednumber theory

Problem Statement

Let P(x,y),Q(x,y)P(x,y),Q(x,y) be two-variable polynomials with the coefficients of integer.Supposed that when an,bna_n,b_n are determined for certain integers a0, b0a_0,\ b_0 by an+1=P(an,bn),  bn+1=Q(an,bn)  (n=0,1,2,⋯ )a_{n+1}=P(a_n,b_n),\ \ b_{n+1}=Q(a_n,b_n)\ \ (n=0,1,2,\cdots) there existed positive integer kk such that (a1,b1)≠(a0,b0)(a_1,b_1)\neq (a_0,b_0) and (ak,bk)=(a0,b0)(a_k,b_k)=(a_0,b_0).Prove that the number of the lattice points on the segment with end points of (an,bn)(a_n,b_n) and (an+1,bn+1)(a_{n+1},b_{n+1}) is indepedent of nn.
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