Legendre Symbol

Related Concepts

Jacobi Symbol; Prime Number; Quadratic Residue

Background

The Legendre symbol was introduced by A.M. Legendre in 1798.

Definition

The Legendre symbol of an integer x modulo a prime p is 0 if x is divisible by p, and otherwise + 1 if x has a square root modulo p, and − 1 if not.

Theory

Let p be an odd prime number and let x be an integer. If x is a quadratic residue, i.e., if x is relatively prime to p and the equation (Modular arithmetic)

$$x \equiv {y}^{2}\mbox{ mod }p$$

has an integer solution y, then the Jacobi symbol of x modulo p, written as (x ∕ p) or $\left (\frac{x} {p}\right )$, is + 1. If x is a quadratic nonresidue – i.e., x is relatively prime to p and has no square roots – then its Legendre symbol is − 1. If x is not relatively prime to p then $\left (\frac{x} {p}\right ) = 0$.

The Legendre symbol may be efficiently computed using the Quadratic Reciprocity Theorem or by modular exponentiation (Exponentiation Algorithms) as

$$\left (\frac{x} {p}\right...