Legendre symbol

Legendre symbols

The Legendre symbol of an integer a over an odd prime p is defined

$\left({a \over p}\right)\cong a^{\frac {p-1}{2}}\mod p;$

$\left({a \over p}\right)\in \{-1,0,1\}.$

Euler’s criterion

By Fermat’s little theorem we have

$a^{p}\cong a\mod p$

and if a is not divisible by the odd prime p, then

$a^{p-1}\cong 1\mod p.$

To be continued ...