Euler's criterion

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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