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4p^2-8p+1|x_i - (2p)^k odd prime p , x_{n+2}= 4x_{n+1}-x_n

Source: 2006 VMEO III Shortlist SL N3 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vme

November 24, 2021
number theorydivides

Problem Statement

Given odd prime pp. Sequence xn{x_n} is defined by xn+2=4xn+1−xnx_{n+2}= 4x_{n+1}-x_n. Choose x0,x1x_0,x_1 such that for every random positive integer kk, there exists i∈Ni\in \mathbb N such that 4p2−8p+1∣xi−(2p)k4p^2-8p+1|x_i - (2p)^k.
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