Question 2
Let $P(x)$ be a non-constant polynomial with integer coefficients such that $P(0) \neq 0$. Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of integers such that $P(i-j)$ divides $a_{i}-a_{j}$ for all distinct positive integers $i, j$. Prove that the sequence $a_{1}, a_{2}, a_{3}, \ldots$ must be constant, that is, $a_{n}$ equals a constant $c$ for all $n$ positive integer.