Quadratic Residue

Related Concepts

Jacobi Symbol; Legendre Symbol; Quadratic Residuosity Problem

Definition

An integer x is a quadratic residue modulo another integer n if x is relatively prime to n and has a square root modulo n.

Theory

Let n be an odd, positive integer, and let x be an integer that is relatively prime to n (Modular arithmetic). The integer x is a quadratic residue modulo n if the equation

$$x \equiv {y}^{2}(\mathrm{mod})n$$

has an integer solution y. In other words, the integer x is a square modulo n. The integer x is a quadratic non-residue otherwise.

If n is an odd prime number, then exactly half of all integers x relatively prime to n are quadratic residues. If n is the product of two distinct odd primes p and q, then the fraction is one-quarter.

Open Problems

Refer Quadratic Residuosity Problem