How do I prove that if $n$ is a quadratic residue of $m > 2$ then $n^{\varphi(m)/2}\equiv 1 \pmod{m}$

Question

Please, give me an idea of how to get started.

Answer 1

You must be using the convention that a quadratic residue mod $m$ is coprime to $m$: your statement is certainly not true if $r$ is not coprime to $m$. If $r$ is such a quadratic residue, $r \equiv x^2 \mod m$ for some $x$, which is also coprime to $m$. Then $r^{\varphi(m)/2} \equiv x^{\varphi(m)} \equiv 1 \mod m$.