ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The four square roots of $9$ modulo $128$

Example

The square roots of $9$ modulo $128$ are

$3, 61, 67, 125.$

Facts & Assumptions

Given: The unit target $9$ modulo $128 = 2^{7}$ .

[L1]

For $k \geq 3$ , an odd target is a square modulo $2^{k}$ exactly when it is $1$ modulo $8$ , and every soluble target has exactly four roots (Unit square criterion and root count modulo powers of two).

Verification

technique · direct

1.1givenalgebra

The representatives are distinct modulo $128$ , and $3^{2} = 9$ , $61^{2} = 29 \cdot 128 + 9$ , $67^{2} = 35 \cdot 128 + 9$ , and $125^{2} = 122 \cdot 128 + 9$ . Thus all four displayed classes are roots.

2.1step 1.1L1∎

Since $9 \equiv 1 (m o d 8)$ , [L1] says that the congruence is soluble and has exactly four root classes. The four distinct roots in step 1.1 are therefore complete.

Depends on

Unit square criterion and root count modulo powers of two

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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.

The four square roots of 99 modulo 128128