Euler's criterion

Again this can be computed using Euler's criterion.

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.

Euler's criterion can be concisely reformulated using the Legendre symbol:

This is Euler's criterion.

Euler's criterion again gives:

This is the desired result, because by Euler's criterion the left hand side is just an alternative expression for the Legendre symbol (a/p).

Two other characterizations of squares modulo a prime are Euler's criterion and Zolotarev's lemma.

In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime.

Euler's criterion is related to the Quadratic reciprocity and is used in a definition of Euler-Jacobi pseudoprimes.

This can be computed using Euler's criterion; This confirms 10 being a square and hence the algorithm can be applied.

Step 2: Take as the quadratic nonresidue (2 is a quadratic nonresidue since (again, Euler's criterion)).