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Definition: Quadratic Residue, Quadratic Nonresidue

Let $m > 0$ be a positive integer. An integer $n\in\mathbb Z$ is called a quadratic residue modulo $m,$ if the congruence $$x^2(m)\equiv n(m)$$ is solvable, otherwise it is called a quadratic nonresidue.

Examples

• For the module $m=1$, any integers $n$ is a residue, since $x^2(1)\equiv 0 (1)\equiv n(1)$ is always solvable.
• If $n=0,$ $n=1,$ or $n$ is a perfect square, then $n$ is a quadratic residue modulo any positive integer $m > 0.$

Mentioned in:

Definitions: 1
Explanations: 2
Problems: 3
Proofs: 4 5 6
Propositions: 7 8 9

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927