In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.

Definition

Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as Legendre's original definition was by means of the explicit formula By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent. Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation aRp, aNp according to whether a is a residue or a non-residue modulo p. For typographical convenience, the Legendre symbol is sometimes written as (a | p) or (a/p). For fixed p, the sequence is periodic with period p and is sometimes called the Legendre sequence. Each row in the following table exhibits periodicity, just as described.

Table of values

The following is a table of values of Legendre symbol with p ≤ 127, a ≤ 30, p odd prime.

Properties of the Legendre symbol

There are a number of useful properties of the Legendre symbol which, together with the law of quadratic reciprocity, can be used to compute it efficiently.

Legendre symbol and quadratic reciprocity

Let p and q be distinct odd primes. Using the Legendre symbol, the quadratic reciprocity law can be stated concisely: Many proofs of quadratic reciprocity are based on Euler's criterion In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.

Related functions

Computational example

The above properties, including the law of quadratic reciprocity, can be used to evaluate any Legendre symbol. For example: Or using a more efficient computation: The article Jacobi symbol has more examples of Legendre symbol manipulation. Since no efficient factorization algorithm is known, but efficient modular exponentiation algorithms are, in general it is more efficient to use Legendre's original definition, e.g. using repeated squaring modulo 331, reducing every value using the modulus after every operation to avoid computation with large integers.