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# Legendre symbol

The Legendre symbol, named after the French mathematician Adrien-Marie Legendre, is used in connection with factorization and quadratic residues.

## Definition

If p is an odd prime number and a is an integer, then the Legendre symbol

$\displaystyle{ \left(\frac{a}{p}\right) }$

is:

• 0 if p divides a;
• 1 if a is a square modulo p — that is to say there exists an integer k such that k2a (mod p), or in other words a is a quadratic residue modulo p;
• −1 if a is not a square modulo p, or in other words a is not a quadratic residue modulo p.

## Properties of the Legendre symbol

There are a number of useful properties of the Legendre symbol which can be used to speed up calculations. They include:

1. $\displaystyle{ \left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right) }$ (it is a completely multiplicative function in its top argument)
2. If ab (mod p), then $\displaystyle{ \left(\frac{a}{p}\right) = \left(\frac{b}{p}\right) }$
3. $\displaystyle{ \left(\frac{1}{p}\right) = 1 }$
4. $\displaystyle{ \left(\frac{-1}{p}\right) = (-1)^{(p-1)/2} }$, i.e. = 1 if p ≡ 1 (mod 4) and = −1 if p ≡ 3 (mod 4)
5. $\displaystyle{ \left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8} }$, i.e. = 1 if p ≡ 1 or 7 (mod 8) and = −1 if p ≡ 3 or 5 (mod 8)
6. If q is an odd prime then $\displaystyle{ \left(\frac{q}{p}\right) = \left(\frac{p}{q}\right)(-1)^{ ((p-1)/2) ((q-1)/2) } }$

The last property is known as the law of quadratic reciprocity. The properties 4 and 5 are traditionally known as the supplements to quadratic reciprocity.

The Legendre symbol is related to Euler's criterion and Euler proved that

$\displaystyle{ \left(\frac{a}{p}\right) \equiv a^{(p-1)/2}\,\pmod p }$