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 Skip Navigation LinksMath Help > Number Theory > Theorems > Quadratic Reciprocity

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 

Numericana, Final Answers: Quadratic Reciprocity 

Related pages in this website

Legendre symbol —  ( a

p
)  , where a is any integer, and p is an odd prime
Jacobi symbol —  ( a

n
)  , where a is any integer, and n is a positive integer greater than 2, an extension of the Legendre symbol.
Kronecker symbol —  ( a

n
)  , where a and n are any integers, an extension of the Jacobi symbol.

Fermat's Little Theorem 

Euler's Totient Theorem 

 

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