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