MathNet 2017-part-06 — 176 problems (EN–VI) Question 13

Let $a$ be a positive integer such that for every positive integer $n$ the number $n^{2} a-1$ has at least one divisor greater than 1 which leaves remainder 1 when divided by $n$. Prove that $a$ is a perfect square.

(Dušan Đukić)