Euler's criterion

Named after Leonhard Euler (1707–1783), Swiss mathematician and physicist, who devised the criterion in 1748.

• IPA^(key): /ˈɔɪ̯.lə(ɹ)z kɹaɪ̯.ˈtɪ.ɹi.ən/, /ˈɔɪ̯.lə(ɹ)z kɹɪ.ˈtɪ.ɹi.ən/

Euler's criterion

1. (number theory) A theorem which states that, given a positive integer a which is coprime to an odd prime number p, a is a quadratic residue of p if and only if ${\displaystyle a^{(p-1)/2}}$ is congruent to 1 modulo p.

• The calculation on the left side of the congruence — or the result thereof — can be denoted with the Legendre symbol.