# The Quadratic Character of $2$

by Rafael Jakimczuk (Universidad Nacional de Luján Argentina)

This article originally appeared in:
Mathematics Magazine
April, 2011

Subject classification(s): Algebra and Number Theory | Number Theory
Applicable Course(s): 4.3 Number Theory

The number $2$ is a quadratic residue mod $p$ if $p = 8k + 1$ or $p = 8k + 7$, but not if $p = 8k + 3$ or $p = 8k + 5$. This is proved by a simple counting argument, assuming the existence of a primitive root mod $p$.

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Capsule Course Topic(s):
Number Theory | Congruences, Solving Congruence Equations