DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Quadratic residues and nonresidues modulo an integer

Definition

Let $n \geq 2$ and let $a \in Z$ satisfy $\gcd (a, n) = 1$ . The integer $a$ is a quadratic residue modulo $n$ if there is an integer $x$ with

$x^{2} \equiv a (m o d n),$

and otherwise it is a quadratic nonresidue modulo $n$ .

By For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ , the coprimality hypothesis says that $[a]_{n}$ is a unit. Thus the terms quadratic residue and quadratic nonresidue here apply only to unit classes; a nonunit target belongs to neither class.

Depends on

Used by

A soluble square congruence need not define a quadratic residue Counterexample
Two quadratic nonresidues modulo 15 can have a nonresidue product Counterexample
The Legendre symbol, including its zero value Definition
Quadratic residuosity is representative-independent and the residues are the image of squaring Proposition

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. Hackman, Elementary Number Theory, Chapter D, Section D.I (standard reference, not scraped)
W. Stein, Elementary Number Theory, Section 4.1 (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 9, Section 1 (standard reference, not scraped)