Quadratic residue
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In mathematics, a number q is called a quadratic residue modulo n if there exists an integer x such that:
- <math>{q}\equiv{x^2}\mbox{ (mod }n\mbox{)}.</math>
Otherwise, q is called a quadratic non-residue. For prime moduli, roughly half of the residue classes are of each type. More precisely, for prime p > 2, there are
- (p − 1)/2
of each kind, excluding 0. They occur in a rather random pattern; this has been exploited in applications to acoustics.
In effect, a quadratic residue modulo n is a number that has a square root in modular arithmetic when the modulus is n. This concept plays a large part in classical number theory.
The problem of finding a square root in modular arithmetic, in other words solving the above for x given q and n, can be a difficult problem. For general composite n, the problem is known to be equivalent to integer factorization of n (an efficient solution to either problem could be used to solve the other efficiently). On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete; however, this is a fixed-parameter tractable problem, where c is the parameter.
See also
- congruence of squares
- distribution of quadratic residues
- Gauss's lemma
- law of quadratic reciprocity
- Legendre symbol
- Paley graph
- quadratic residuosity problem
- Zolotarev's lemma
External links
- MathWorld: Quadratic Residue
- Template:Cite book A7.1: AN1, pg.249.de:Quadratischer Rest
es:residuo cuadrático fr:Résidu quadratique he:שארית ריבועית ja:平方剰余 pl:Reszta kwadratowa fi:Neliöjäännös sv:Kvadratisk rest zh:二次剩余