# Difference between revisions of "Euler's criterion"

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where the expression on the right side is the [[Legendre symbol]], defined to be <math>+1</math> for a [[quadratic residue]] and <math>-1</math> for a [[quadratic nonresidue]]. Note that the Legendre symbol is the restriction to primes of the [[Jacobi symbol]]. | where the expression on the right side is the [[Legendre symbol]], defined to be <math>+1</math> for a [[quadratic residue]] and <math>-1</math> for a [[quadratic nonresidue]]. Note that the Legendre symbol is the restriction to primes of the [[Jacobi symbol]]. | ||

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | ===Applications=== | ||

+ | |||

+ | * [[Congruence condition for minus one to be a quadratic residue]] | ||

+ | * [[Congruence condition for two to be a quadratic residue]] | ||

+ | * [[Quadratic reciprocity]] | ||

+ | |||

+ | ===Primality tests=== | ||

+ | |||

+ | * [[Euler primality test]], which is not conclusive and can be fooled by [[Euler pseudoprime]]s to the given base | ||

+ | * [[Euler-Jacobi primality test]], which is not conclusive and can be fooled by [[Euler-Jacobi pseudoprime]]s to the given base |

## Latest revision as of 20:19, 2 January 2012

## Contents

## Statement

### In terms of quadratic residues and nonresidues

Suppose is an odd prime number. Consider an integer that is not zero mod . Then:

- is congruent to either 1 or -1 mod .
- is congruent to 1 mod if and only if is a quadratic residue mod .
- is congruent to -1 mod if and only if is a quadratic nonresidue mod .

### In terms of Legendre symbol

Suppose is an odd prime number. Consider an integer that is not zero mod . Then:

where the expression on the right side is the Legendre symbol, defined to be for a quadratic residue and for a quadratic nonresidue. Note that the Legendre symbol is the restriction to primes of the Jacobi symbol.

## Related facts

### Applications

- Congruence condition for minus one to be a quadratic residue
- Congruence condition for two to be a quadratic residue
- Quadratic reciprocity

### Primality tests

- Euler primality test, which is not conclusive and can be fooled by Euler pseudoprimes to the given base
- Euler-Jacobi primality test, which is not conclusive and can be fooled by Euler-Jacobi pseudoprimes to the given base