#### Definition 16.3.1.

Assume that \(a\not\equiv 0\) (mod \(p\)), for \(p\) a prime.

- If there
*is*a solution of \(x^2\equiv a\) (mod \(p\)) we say that \(a\) is a quadratic residue of \(p\) (or a QR). - If there
*is not*a solution of \(x^2\equiv a\) (mod \(p\)) we say that \(a\) is a quadratic nonresidue of \(p\text{.}\)

Although some authors also define this notion for composite moduli (as does Sage, see Sage note 16.3.3), we will go with the majority and reserve these terms for prime moduli.