Legendre Symbol - Definition

Definition

Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as follows:

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

Legendre's original definition was by means of an explicit formula:

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent. Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation, according to whether a is a residue or a non-residue modulo p.

For typographical convenience, the Legendre symbol is sometimes written as (a|p) or (a/p). The sequence (a|p) for a equal to 0,1,2,... is periodic with period p and is sometimes called the Legendre sequence, with {0,1,−1} values occasionally replaced by {1,0,1} or {0,1,0}.

Read more about this topic:  Legendre Symbol

Famous quotes containing the word definition:

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)