Legendre Symbol
The Legendre symbol is a number theoretic function 
which is defined
to be equal to 
depending on whether 
is a quadratic
residue modulo 
. The definition is sometimes generalized
to have value 0 if 
,
 |
(1)
|
If 
is an odd prime,
then the Jacobi symbol reduces to the Legendre symbol.
The Legendre symbol is implemented in the Wolfram
Language via the Jacobi symbol, JacobiSymbol[a,
p].
The Legendre symbol obeys the identity
 |
(2)
|
Particular identities include
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(Nagell 1951, p. 144), as well as the general
![(q/p)=(p/q)(-1)^([(p-1)/2][(q-1)/2])](/images/equations/LegendreSymbol/NumberedEquation3.gif) |
(7)
|
when 
and 
are both odd
primes.
In general,
 |
(8)
|
if 
is an odd prime.
Contact the MathWorld Team
legendre symbol
