Euler's Criterion

Theorem

Let $a$ be a residue order $n$ of $m$, where $a$ and $m$ are coprime.

Then:

$a^{\map \phi m / d} \equiv 1 \pmod m$

where:

$\map \phi m$ denotes the Euler $\phi$ function of $m$

$d$ denotes the gretest common divisor of $\map \phi m$ and $n$

$\equiv$ denotes modulo congruence.

Euler's Criterion for Quadratic Residue

Let $p$ be an odd prime.

Let $a \not \equiv 0 \pmod p$.

Then:

$\ds a^{\frac {p-1} 2}$

$\equiv$

$\ds 1$

$\ds \pmod p$

if and only if $a$ is a quadratic residue of $p$

$\ds a ^{\frac {p-1} 2}$

$\equiv$

$\ds -1$

$\ds \pmod p$

if and only if $a$ is a quadratic non-residue of $p$.

Proof

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Source of Name

This entry was named for Leonhard Paul Euler.

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's criterion
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): residue: 2.
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's criterion
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): residue: 2.