Legendre symbol

Etymology
Named after French mathematician (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)  A mathematical function of an integer and a prime number, written $$\left({a \over p}\right)$$, which indicates whether a is a quadratic residue modulo p.
 * 2) * 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 $$\textstyle\frac{p-1}{2}$$ congruences (unless, of course, we are fortunate enough to encounter the desired quadratic residue along the way).

Usage notes
The symbol takes the values:

$$\left(\frac{a}{p}\right) = \begin{cases} -1 & \text{ if } a \text{ is a quadratic nonresidue modulo } p, \\ 0 & \text{ if } a \equiv 0 \pmod{p}, \\ 1 & \text{ if } a \text{ is a quadratic residue modulo } p \text{ and } a \not\equiv 0\pmod{p}. \end{cases}$$

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.