Legendre symbol (a/p) for various a (along top) and p (along left side).
a
p
0
1
2
3
4
5
6
7
8
9
10
3
0
1
−1
5
0
1
−1
−1
1
7
0
1
1
−1
1
−1
−1
11
−0
−1
−1
1
−1
1
−1
−1
−1
−1
−1
Only 0 ≤ a < p are shown, since due to the first property below any other a can be reduced modulo p. Quadratic residues are highlighted in yellow, and correspond precisely to the values 0 and 1.
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[1] 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.
^Legendre, A. M. (1798). Essai sur la théorie des nombres. Paris. p. 186.
In number theory, the Legendresymbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p:...
The Jacobi symbol is a generalization of the Legendresymbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other...
reciprocity — Let p and q be distinct odd prime numbers, and define the Legendresymbol as: ( q p ) = { 1 if n 2 ≡ q mod p for some integer n − 1 otherwise...
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sum, for R the field of residues modulo a prime number p, and χ the Legendresymbol. In this case Gauss proved that G(χ) = p1⁄2 or ip1⁄2 for p congruent...
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