Quadratic residue

English

Noun

1. Template:Lb For given positive integer n, any integer that is congruent to some square m² modulo n.
   • Template:Quote-book
   • 1991, John Stillwell (translator), Template:W, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], Template:W, Template:W, 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 $\equiv$ 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, Template:W [清华大学出版社有限公司], 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.

Usage notes

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

Antonyms

• Template:Antsense Template:L

Translations

• Danish: Template:T
• Italian: Template:T