Euler's criterion

== English == === Etymology === Named after Leonhard Euler (1707–1783), Swiss mathematician and physicist, who devised the criterion in 1748. === Pronunciation === IPA(key): /ˈɔɪ̯.lə(ɹ)z kɹaɪ̯ˈtɪ.ɹi.ən/, /ˈɔɪ̯.lə(ɹ)z kɹɪˈtɪ.ɹi.ən/ === Proper noun === Euler's criterion (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 a ( p − 1 ) / 2 {\displaystyle a^{(p-1)/2}} is congruent to 1 modulo p. ==== Usage notes ==== The calculation on the left side of the congruence — or the result thereof — can be denoted with the Legendre symbol.