VN-TST 1997 Problem 4

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