Problem

Source: Mongolia TST 2011 Test 4 #3

Tags: modular arithmetic, algebra, polynomial, Vieta, number theory unsolved, number theory



Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)