Problem

Source: Switzerland - 2017 Swiss MO Final Round p7

Tags: number theory, Perfect Square, diophantine, Diophantine equation



Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$