Quadratic residue

Definition

Let $n$ be a positive integer and let $q$ be an integer. Then $q$ is a quadratic residue modulo $n$ if there is an integer $x$ with $x^2 \equiv q \pmod n$ . Otherwise $q$ is a quadratic nonresidue modulo $n$ .

Remarks

Many texts also require $\gcd(q, n) = 1$ , and so they do not count $0$ as a quadratic residue. Modulo $7$ , the squares of $1, \dots, 6$ are $1, 4, 2, 2, 4, 1$ . So the nonzero quadratic residues modulo $7$ are $1$ , $2$ and $4$ . Modulo an odd prime $p$ , exactly $(p - 1)/2$ of the classes $1, \dots, p - 1$ are quadratic residues. The product of two quadratic residues is a quadratic residue.

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Divisibility definition
Congruence modulo n definition
Quadratic residue definition

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18 pages build on this one.

Tag	`0SQDB87Z3T`
URI	`m2://0SQDB87Z3T`
Aliases	`quadratic-nonresidue`
Kind	definition
Writing	W1 stated
MSC	11A15
Defines	Quadratic residue
Uses	Congruence modulo n

Quadratic residue

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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