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
(Nagell 1951, p. 144), as well as the general
(7)
when
and
are both odd primes .
In general,
(8)
if
is an odd prime .
See also Jacobi Symbol ,
Kronecker Symbol ,
Quadratic Reciprocity Theorem ,
Quadratic Residue
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References Guy, R. K. "Quadratic Residues. Schur's Conjecture." §F5 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 244-245,
1994. Hardy, G. H. and Wright, E. M. "Quadratic Residues."
§6.5 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 67-68, 1979. Jones, G. A. and Jones, J. M.
"The Legendre Symbol." §7.3 in Elementary
Number Theory. Berlin: Springer-Verlag, pp. 123-129, 1998. Nagell,
T. "Euler's Criterion and Legendre's Symbol." §38 in Introduction
to Number Theory. New York: Wiley, pp. 133-136, 1951. Shanks,
D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 33-34
and 40-42, 1993. Referenced on Wolfram|Alpha Legendre Symbol
Cite this as:
Weisstein, Eric W. "Legendre Symbol."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LegendreSymbol.html
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