MathDB
Arbitrary integers not smaller than k

Source: Vietnam TST 1997, Problem 4

July 28, 2008
functionalgebrapolynomialinductionnumber theoryrelatively primenumber theory unsolved

Problem Statement

The function f:N→Z f : \mathbb{N} \to \mathbb{Z} is defined by f(0) \equal{} 2, f(1) \equal{} 503 and f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n) for all n∈N n \in\mathbb{N}. Let s1 s_1, s2 s_2, … \ldots, sk s_k be arbitrary integers not smaller than k k, and let p(si) p(s_i) be an arbitrary prime divisor of f(2si) f\left(2^{s_i}\right), ( i \equal{} 1, 2, \ldots, k). Prove that, for any positive integer t t (t≤k t\le k), we have 2^t \Big | \sum_{i \equal{} 1}^kp(s_i) if and only if 2t∣k 2^t | k.
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