View source for Congruence condition for two to be a quadratic residue

==Statement==

Suppose <math>p</math> is an odd prime. Then:

* <math>2</math> is a [[fact about::quadratic residue]] modulo <math>p</math> if and only if <math>p \equiv \pm 1 \pmod 8</math>, i.e., <math>p \equiv 1 \pmod 8</math> or <math>p \equiv 7 \pmod 8</math>.
* <math>2</math> is a [[fact about::quadratic nonresidue]] modulo <math>p</math> if and only if <math>p \equiv \pm 3 \pmod 8</matH>, i.e, <math>p \equiv 3 \pmod 8</math> or <math>p \equiv 5 \pmod 8</math>.

==Related facts==

===Similar facts===
* [[Congruence condition for minus one to be a quadratic residue]]
* [[Quadratic reciprocity]]

===Applications===

* [[Every integer that is not a perfect square is a quadratic nonresidue for infinitely many primes]]
* [[Every integer is a quadratic residue for infinitely many primes]]
* [[Safe prime has plus or minus two as a primitive root]]