Noun
Legendre symbol (plural Legendre symbols)
(number theory) A mathematical function of an integer and a prime number, written
(
a
p
)
{\displaystyle \left({a \over p}\right)}
, which indicates whether a is a quadratic residue modulo p.
For example: : Or using a more efficient computation: : The article Jacobi symbol has more examples of Legendre symbol manipulation. Source: Internet
Gauss's cases 9) - 14) above can be expressed in terms of Jacobi symbols: : and since p is prime the left hand side is a Legendre symbol, and we know whether M is a residue modulo p or not. Source: Internet
Keep in mind that Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol. Source: Internet
Legendre symbol and quadratic reciprocity Let p and q be odd primes. Source: Internet
Legendre symbol (a/p) for various a (along top) and p (along left side). Source: Internet
Related functions * The Jacobi symbol is a generalization of the Legendre symbol that allows for a composite second (bottom) argument n, although n must still be odd and positive. Source: Internet