Quadratic residue

In modular arithmetic, a quadratic residue for the modulus N is a number which can be expressed as the residue of a2 modulo N for some integer a. A quadratic non-residue of N is a number which is not a quadratic residue of N.

Legendre symbol
When the modulus is a prime p, the Legendre symbol $$\left(\frac{a}{p}\right)$$ expresses the quadratic nature of a modulo p. We write


 * $$\left(\frac{a}{p}\right) = 0 \, $$ if p divides a;
 * $$\left(\frac{a}{p}\right) = +1 \, $$ if a is a quadratic residue of p;
 * $$\left(\frac{a}{p}\right) = -1 \, $$ if a is a quadratic non-residue of p.

The Legendre symbol is multiplicative, that is,


 * $$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right) . \, $$