Legendre symbol

English

Etymology

Named after French mathematician Template:W (1752–1833), who introduced the symbol in 1798 in his work Essai sur la Théorie des Nombres ("Essay on the Theory of Numbers").

Noun

1. Template:Lb A mathematical function of an integer and a prime number, written $\left(\frac{a}{p}\right)$, which indicates whether a is a quadratic residue modulo p.
   • 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 $\frac{p-1}{2}$ congruences (unless, of course, we are fortunate enough to encounter the desired quadratic residue along the way).

   • Template:Quote-book
   • Template:Quote-book

Usage notes

The symbol takes the values:

$\left(\frac{a}{p}\right)=\{\begin{array}{cc}-1& \text{ if }a\text{ is a quadratic nonresidue modulo }p,\\ 0& \text{ if }a\equiv 0(\mathrm{mod}p),\\ 1& \text{ if }a\text{ is a quadratic residue modulo }p\text{ and }a\equiv ̸0(\mathrm{mod}p).\end{array}$

It is generalised to composite numbers by the Jacobi symbol, which is identical in form and range of values and is defined as a product of Legendre symbols.

See also

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