PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The Legendre symbol is well defined on residue classes

Statement

For every odd prime $p$ , the Legendre symbol belongs to ${- 1, 0, 1}$ , depends only on the numerator modulo $p$ , and satisfies

$(\frac{a}{p}) = 0 ⟺ p ∣ a .$

Facts & Assumptions

Given: An odd prime $p$ and integers $a, b$ with $a \equiv b (m o d p)$ .

[L1]

The Legendre symbol is $0$ on a numerator divisible by $p$ , $1$ on a quadratic residue modulo $p$ , and $- 1$ on a quadratic nonresidue (The Legendre symbol, including its zero value).

[L2]

The congruence $a \equiv b (m o d p)$ means that $p$ divides $a - b$ (Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ ).

[L3]

Quadratic residuosity of a unit integer depends only on its residue class (Quadratic residuosity is representative-independent and the residues are the image of squaring).

Proof

technique · direct

1.1L1L2given

By [L2], $p ∣ (a - b)$ , so $p ∣ a$ exactly when $p ∣ b$ . Thus congruent numerators enter the zero branch of [L1] simultaneously.

2.1L1L3step 1.1

If $p ∤ a$ , then also $p ∤ b$ by step 1.1, and [L3] says that $a$ and $b$ are simultaneously quadratic residues or simultaneously nonresidues. Hence [L1] assigns them the same sign.

3.1L1step 1.1step 2.1∎

The three disjoint branches in [L1] give only the values $- 1, 0, 1$ ; step 1.1 proves that divisibility gives value zero, and the two unit branches give nonzero values. Therefore the symbol is representative-independent and is zero exactly when $p$ divides its numerator.

Depends on

Used by

Dependency tree · two levels

9 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)
A. Gorodnik, Number Theory, Lecture 9, Section 1 (standard reference, not scraped)