Difference between revisions of "Euler's criterion"

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* [[Congruence condition for two to be a quadratic residue]]
 
* [[Congruence condition for two to be a quadratic residue]]
 
* [[Quadratic reciprocity]]
 
* [[Quadratic reciprocity]]
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===Primality tests===
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* [[Euler primality test]], which is not conclusive and can be fooled by [[Euler pseudoprime]]s to the given base
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* [[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

Statement

In terms of quadratic residues and nonresidues

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

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

Primality tests