Prove that there exists $s$ such that $s(ab-1)^n +1$ is composite I find this interesting question in a number theory book.
Given two positive integers $a, b$ such that $a>1, b>1, \gcd(a, b)=1$. Prove that there exists a positive integer $s$ such that $s(ab-1)^n + 1$ is a composite number $\forall n \in \mathbb{N}.$
Its solution is short but beautiful :)

Bill's hint is more than enough and kills it beautifully, but as another solution, let $p$ be a prime divisor of $ab-2$ and take $s = (p-1)!.$ By Wilson's theorem, we'd have
$$(p-1)!(ab-1)^n + 1 \equiv (-1)(1)^n + 1 \equiv 0 \;\; (\text{mod} \; p)$$
P.S. Coprime condition is redundant.
