DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The Legendre symbol, including its zero value

Definition

Let $p$ be an odd prime and let $a \in Z$ . The Legendre symbol is

$(\frac{a}{p}) : = {\begin{cases} 0, & p ∣ a, \\ 1, & p ∤ a and a is a quadratic residue modulo p, \\ - 1, & p ∤ a and a is a quadratic nonresidue modulo p . \end{cases}$

The zero branch is separate from the quadratic residue and nonresidue branches of Quadratic residues and nonresidues modulo an integer, which apply only to unit numerators.

Depends on

Quadratic residues and nonresidues modulo an integer

Used by

x²≡ a pmod p has exactly 1+(a/p) solution classes Corollary
A soluble square congruence need not define a quadratic residue Counterexample
A nonmonic quadratic congruence solved through its discriminant Example
Quadratic residues, roots, and the Legendre table modulo 11 Example
On the units, the Legendre symbol is the unique nontrivial homomorphism to {±1} Proposition
The Legendre symbol is well defined on residue classes Proposition
Euler's criterion: (a/p)≡ a^(p-1)/2pmod p Theorem
First supplement: (-1/p)=(-1)^(p-1)/2 Theorem
Gauss's quadratic-residue lemma Theorem
The Legendre symbol is multiplicative for all integer numerators Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)