Editing Quadratic residue (section)

{{short description|Integer that is a perfect square modulo some integer}}
{{pp-sock|small=yes}}
In [[number theory]], an [[integer]] ''q'' is a '''quadratic residue''' [[modulo operation|modulo]] ''n'' if it is [[Congruence relation|congruent]] to a [[Square number|perfect square]] modulo ''n''; that is, if there exists an integer ''x'' such that

:<math>x^2\equiv q \pmod{n}.</math>

Otherwise, ''q'' is a  '''quadratic nonresidue''' modulo ''n''.

Quadratic residues are used in applications ranging from [[acoustical engineering]] to [[cryptography]] and the [[Integer factorization|factoring of large numbers]].