quadratic residue

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quadratic residue

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quadratic residue (plural quadratic residues)

1. (number theory, modular arithmetic) For given positive integer n, any integer that is congruent to some square m² modulo n.
   • 1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52:

     There are many small tables of quadratic residues giving for the first few primes p the positive quadratic residues of p arranged in order of their size.

   • 1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, , American Mathematical Society, London Mathematical Society, page 82,

     If we now make the assumption that q is a quadratic residue of all odd primes z not greater than 2m + 1, then it follows from earlier theorems (§37) that the prime q, since it is ${\displaystyle \equiv }$ 1 (mod 8) and hence a quadratic residue of each power of 2, is also a quadratic residue of each number which has no odd prime factors except the prime numbers z

   • 1999, Dinakar Ramakrishnan, Robert J. Valenza, Tsinghua University Press , Fourier Analysis on Number Fields, page 213,

     Of special importance here is the quadratic reciprocity law, which for primes p and q gives a precise relationship between the status of p as a quadratic residue mod q and the status of q as a quadratic residue mod p.

An integer satisfying the criterion is said to be a quadratic residue modulo n. The trivial case m = 0 is usually excluded.

• (antonym(s) of “integer expressible as a square modulo n”): quadratic nonresidue