Meaning of Legendre symbol | Babel Free

Definitions

A mathematical function of an integer and a prime number, written (a/p), which indicates whether a is a quadratic residue modulo p.

Examples

“1994, James K. Strayer, Elementary Number Theory, Waveland Press, 2002, Reissue, page 109, Our only method at present for the computation of Legendre symbols requires a possible consideration of (p-1)/2 congruences (unless, of course, we are fortunate enough to encounter the desired quadratic residue along the way).”

“The Jacobi symbol, which generalizes the Legendre symbol, sheds some additional light on how to determine whether (7.29) has solutions when m has two or more distinct prime factors.”

“Jacobi symbols can be used to facilitate the calculation of Legendre symbols. In fact, Legendre symbols can be eventually calculated by Jacobi symbols [17]. That is, the Legendre symbol can be calculated as if it were a Jacobi symbol. For example, consider the Legendre symbol #92;left(#92;frac#123;335#125;#123;2999#125;#92;right) where 335 = 5·67 is not a prime (of course, 2999 is a prime, otherwise, it is not a Legendre symbol).”

CEFR level