CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Jacobi symbol minus one obstructs quadratic residuosity

Statement

Let $n$ be an odd positive integer and let $a \in Z$ satisfy $\gcd (a, n) = 1$ . If $(\frac{a}{n}) = - 1$ , then $a$ is not a square modulo $n$ .

Facts & Assumptions

Given: An odd positive integer $n$ and an integer $a$ with $\gcd (a, n) = 1$ and $(\frac{a}{n}) = - 1$ .

[L1]

If $a$ is a unit square modulo an odd positive integer $n$ , then $(\frac{a}{n}) = 1$ (A unit square modulo an odd integer has Jacobi symbol one).

Proof

technique · direct

1.1L1given

If $a$ were a square modulo $n$ , the gcd hypothesis would make it a unit square and [L1] would give $(\frac{a}{n}) = 1$ .

2.1step 1.1given∎

This contradicts the given value $(\frac{a}{n}) = - 1$ , so $a$ is not a square modulo $n$ .

Depends on

A unit square modulo an odd integer has Jacobi symbol one

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

A. Gorodnik, Number Theory, Lecture 10, §1 (standard reference, not scraped)