Little extension of Euler's criterion for quadratic residue (Q2107681)

The main result of this article is the criterion that an integer \(a\) is a quadratic residue modulo \(n\) if \(a^{\rho(n)} \equiv 1 \bmod n\), where \(\rho\) is a certain number theoretic function. This is, however, a special case of the observation that if \(a^m \equiv 1 \bmod n\) for some odd integer \(m = 2k+1\), then \(a\) is a quadratic residue modulo \(n\) since \(a \equiv (a^{-k})^2 \bmod n\).