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,

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

References

• G. H. Hardy; E. M. Wright (2008). An Introduction to the Theory of Numbers, 6th ed. Oxford University Press. ISBN 0-19-921986-9.