ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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, roots, and the Legendre table modulo $11$

Example

Modulo $11$ , the nonzero quadratic residues are $1, 3, 4, 5, 9$ , with root pairs

$1 : {1, 10}, 3 : {5, 6}, 4 : {2, 9}, 5 : {4, 7}, 9 : {3, 8} .$

For representatives $a = 0, 1, \dots, 10$ , the Legendre values are

$0, 1, - 1, 1, 1, 1, - 1, - 1, - 1, 1, - 1.$

Facts & Assumptions

Given: The odd prime modulus $11$ .

[L1]

For an odd prime $p$ , exactly $(p - 1) / 2$ nonzero classes are quadratic residues, the same number are nonresidues, and every nonzero residue has exactly two square roots (An odd prime has $(p - 1) / 2$ nonzero quadratic residues and as many nonresidues).

[L2]

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

Verification

technique · direct

1.1L1givenalgebra

Squaring $1, 2, 3, 4, 5$ modulo $11$ gives $1, 4, 9, 5, 3$ . Their negatives give the same squares, so the displayed pairs are roots of those five distinct classes; [L1] shows that this exhausts all nonzero residues and all roots.

2.1L2step 1.1

The remaining nonzero classes $2, 6, 7, 8, 10$ are therefore nonresidues, while the zero class is in the separate divisible branch of [L2].

3.1L2step 1.1step 2.1∎

Assigning $1$ to the five residue classes, $- 1$ to the five nonresidue classes, and $0$ to the zero class by [L2] gives the displayed table.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

W. Stein, Elementary Number Theory, Example 4.2.4 (standard reference, not scraped)

Quadratic residues, roots, and the Legendre table modulo 1111