Legendre Symbol
المؤلف:
Hardy, G. H. and Wright, E. M.
المصدر:
"Quadratic Residues." §6.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press
الجزء والصفحة:
...
23-8-2020
1432
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 
,
![(a/p)=(a|p)=<span style=]() {0 if p|a; 1 if a is a quadratic residue modulo p; -1 if a is a quadratic nonresidue modulo p. " src="https://mathworld.wolfram.com/images/equations/LegendreSymbol/NumberedEquation1.gif" style="height:64px; width:360px" /> |
(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
![(q/p)=(p/q)(-1)^([(p-1)/2][(q-1)/2])](https://mathworld.wolfram.com/images/equations/LegendreSymbol/NumberedEquation3.gif) |
(7)
|
when 
and 
are both odd primes.
In general,
 |
(8)
|
if 
is an odd prime.
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.
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