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Difference between revisions of "Quadratic residue"
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| − | In [[mathematics]], a number | + | In [[mathematics]], a number {{V|q}} is called a '''quadratic residue''' [[modular arithmetic|modulo]] {{V|p}} if there exists an [[integer]] {{V|x}} such that: |
| − | :<math>{x^2}\equiv{q}\ | + | :<math>{x^2}\equiv{q}\ (mod\ p)</math> |
| − | Otherwise, | + | Otherwise, {{V|q}} is called a '''quadratic non-residue'''. |
| − | In effect, a quadratic residue modulo | + | In effect, a quadratic residue modulo {{V|p}} is a number that has a [[Modular square root|square root]] in [[modular arithmetic]] when the modulus is {{V|p}}. The [[law of quadratic reciprocity]] says something about quadratic residues and [[prime]]s. |
Quadratic residues are used in the [[Legendre symbol]]. [[Quadratic reciprocity]] and the [[Gauss lemma]] both reason about quadratic residues. | Quadratic residues are used in the [[Legendre symbol]]. [[Quadratic reciprocity]] and the [[Gauss lemma]] both reason about quadratic residues. | ||
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==External links== | ==External links== | ||
*[http://mathworld.wolfram.com/QuadraticResidue.html MathWorld: Quadratic Residue] | *[http://mathworld.wolfram.com/QuadraticResidue.html MathWorld: Quadratic Residue] | ||
| − | *[ | + | *[[Wikipedia:Quadratic_residue|Wikipedia]] |
[[Category:Math]] | [[Category:Math]] | ||
Latest revision as of 20:11, 26 October 2020
In mathematics, a number q is called a quadratic residue modulo p if there exists an integer x such that:
- [math]\displaystyle{ {x^2}\equiv{q}\ (mod\ p) }[/math]
Otherwise, q is called a quadratic non-residue.
In effect, a quadratic residue modulo p is a number that has a square root in modular arithmetic when the modulus is p. The law of quadratic reciprocity says something about quadratic residues and primes.
Quadratic residues are used in the Legendre symbol. Quadratic reciprocity and the Gauss lemma both reason about quadratic residues.
