Products of quadratic residues and related identities

Abstract

We study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv 5\,({\rm mod}\, 8)$, then $$\prod _{0 \lt x \lt p/2,\,(\frac {x}{p})=1}x\equ