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 Math Help > Number Theory > Theorems > Quadratic Reciprocity

 If p,q are odd primes, then ( p q )( q p ) = (-1)(p-1)/2(q-1)/2, where
 ( p q ) represents the Legendre symbol.

Another way of phrasing this, which is perhaps more useful, is that

 ( p q ) = ( q p ) unless both p and q are primes of the form 4k+3

### Calculating the value of a Legendre symbol using Quadratic Reciprocity

Example, using Quadratic Reciprocity,

 ( 3 983 ) = (-1) ( 983 3 ) because both 3 and 983 are primes of the form 4k+3
 = (-1) ( 2 3 ) = (-1)(-1) = 1
 More generally, Quadratic Reciprocity can be used to find ( 3 p ) as follows:
 ( 3 p ) = ( p 3 ) if p≡1 (mod 4), and ( 3 p ) = (-1) ( p 3 ) if p≡-1 (mod 4).

Since 1 is the only quadratic residue (mod 3), and -1 is the only quadratic non-residue (mod 3), it follows that

 ( 3 p ) = { 1 if p≡±1 (mod 12),-1 if p≡�5 (mod 12),

### Internet references

Wikipedia:  Law of quadratic reciprocity