Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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∈Z satisfy gcd⁡(a,n)=1. If (an)=−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 (an)=−1.

[L1]

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

Proof

technique · direct
1.1

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

L1given
2.1

This contradicts the given value (an)=−1, so a is not a square modulo n.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 6 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