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# Legendre symbol

Suppose *p* is an odd prime and *a* is any integer. The Legendre symbol (*a*|*p*) is defined to be

+1 if ais a quadratic residue (modp),−1 if ais a quadratic non-residue (modp) and,0 if pdividesa

Note: the Legendre symbol is often written vertically: .

Euler showed that (*a*|*p*) ≡ *a*^{(p−1)/2} (mod *p*). Using this we can
show the following: Let *p* and *q* be odd primes, then

- (−1|
*p*) = 1 if*p*≡ 1 (mod 4), and (−1|*p*) = −1 if*p*≡ 3 (mod 4); - (
*a*|*p*) (*b*|*p*) = (*ab*|*p*); - if
*a*≡*b*(mod*p*), then (*a*|*p*) = (*b*|*p*); - (
*a*^{2}|*p*) = 1 unless*p*divides*a*.

For the prime 2 we have

- (2|
*p*) = 1 if*p*≡ 1 or 7 (mod 8), and

- (2|
*p*) = −1 if*p*≡ 3 or 5 (mod 8).

Far more difficult to prove is the quadratic reciprocity law:

- If
*p*and*q*are distinct primes, then .

In other words, (*p*|*q*) = (*q*|*p*), unless *p* ≡ *q* ≡ 3 (mod 4), in which case (*p*|*q*) = −(*q*|*p*).

The Legendre symbol is often evaluated by using the Jacobi symbol.

**See Also:** JacobiSymbol

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