In number theory, an integer *q* is called a **quadratic residue** modulo *n* if it is congruent to a perfect square modulo *n*; i.e., if there exists an integer *x* such that:

Otherwise, *q* is called a **quadratic nonresidue** modulo *n*.

Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

Read more about Quadratic Residue: History, Conventions, and Elementary Facts, Notations, Distribution of Quadratic Residues, Complexity of Finding Square Roots, The Number of Quadratic Residues, Table of Quadratic Residues

### Famous quotes containing the word residue:

“Every poem of value must have a *residue* [of language].... It cannot be exhausted because our lives are not long enough to do so. Indeed, in the greatest poetry, the *residue* may seem to increase as our experience increases—that is, as we become more sensitive to the particular ignitions in its language. We return to a poem not because of its symbolic [or sociological] value, but because of the waste, or subversion, or difficulty, or consolation of its provision.”

—William Logan, U.S. educator. “Condition of the Individual Talent,” The Sewanee Review, p. 93, Winter 1994.