Editing Legendre symbol (section)

== Definition ==
Let <math>p</math> be an odd [[prime number]]. An integer <math>a</math> is a [[quadratic residue]] modulo <math>p</math> if it is [[modular arithmetic|congruent]] to a [[square number|perfect square]] modulo <math>p</math> and is a quadratic nonresidue modulo <math>p</math> otherwise. The '''Legendre symbol''' is a function of <math>a</math> and <math>p</math> defined as
:<math>\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 nonresidue modulo } p, \\
 0 & \text{if } a \equiv 0 \pmod p.
\end{cases}</math>

Legendre's original definition was by means of the explicit formula
:<math> \left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \pmod p \quad \text{ and } \quad\left(\frac{a}{p}\right) \in \{-1,0,1\}. </math>

By [[Euler's criterion]], which had been discovered earlier and was known to Legendre, these two definitions are equivalent.<ref>Hardy & Wright, Thm. 83.</ref> Thus Legendre's contribution lay in introducing a convenient ''notation'' that recorded quadratic residuosity of ''a''&nbsp;mod&nbsp;''p''. For the sake of comparison, [[Carl Friedrich Gauss|Gauss]] used the notation ''a''R''p'', ''a''N''p'' according to whether ''a'' is a residue or a non-residue modulo ''p''. For typographical convenience, the Legendre symbol is sometimes written as (''a''&nbsp;|&nbsp;''p'') or (''a''/''p'').  For fixed ''p'', the sequence <math>\left(\tfrac{0}{p}\right),\left(\tfrac{1}{p}\right),\left(\tfrac{2}{p}\right),\ldots</math>  is [[periodic sequence|periodic]] with period ''p'' and is sometimes called the '''Legendre sequence'''.  Each row in the following table exhibits periodicity, just as described.