ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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

1.1

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.

L1givenalgebra

2.1

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

L2step 1.1

3.1

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.

L2step 1.1step 2.1∎

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: 50 results over 15 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

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

Quadratic residues, roots, and the Legendre table modulo 1111