quadratic residue

plural quadratic residues

1. modular arithmeticmathematicsFor a given positive integer n, any integer that is congruent to some square m² modulo n.

   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.

   1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 52

   2 more examples

   1991, John Stillwell (translator), Peter Gustav Lejeune Dirichlet, Lectures on Number Theory, [1863, Vorlesungen über Zahlentheorie], 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 ≡ 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 [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.