Euler's criterion
In number theory concerning primes
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {\displaystyle a^{\tfrac {p-1}{2}}\equiv {\begin{cases}\;\;\,1{\pmod {p}}&{\text{ if there is an integer }}x{\text{ such that }}x^{2}\equiv a{\pmod {p}},\\-1{\pmod {p}}&{\text{ if there is no such integer.}}\end{cases}}} Euler's cr...
Nº Q2346904 ★
Common · Knowledge
Euler's criterion
In number theory concerning primes
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {\displaystyle a^{\tfrac {p-1}{2}}\equiv {\begin{cases}\;\;\,1{\pmod {p}}&{\text{ if there is an integer }}x{\text{ such that }}x^{2}\equiv a{\pmod {p}},\\-1{\pmod {p}}&{\text{ if there is no such integer.}}\end{cases}}} Euler's cr...
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From Wikipedia
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p − 1 2 ≡ { 1 ( mod p ) if there is an integer x such that x 2 ≡ a ( mod p ) , − 1 ( mod p ) if there is no such integer. {\displaystyle a^{\tfrac {p-1}{2}}\equiv {\begin{cases}\;\;\,1{\pmod {p}}&{\text{ if there is an integer }}x{\text{ such that }}x^{2}\equiv a{\pmod {p}},\\-1{\pmod {p}}&{\text{ if there is no such integer.}}\end{cases}}} Euler's criterion can be concisely reformulated using the Legendre symbol: ( a p ) ≡ a p − 1 2 ( mod p ) . {\displaystyle \left({\frac {a}{p}}\right)\equiv a^{\tfrac {p-1}{2}}{\pmod {p}}.} The criterion dates from a 1748 paper by Leonhard Euler.
Text: Wikipédia, CC BY-SA 4.0. ·
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