ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-17

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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.

The square roots of $1$ modulo $360$ by the Chinese remainder theorem

Example

The square roots of $1$ modulo $360$ are

$1, 19, 71, 89, 91, 109, 161, 179, 181, 199, 251, 269, 271, 289, 341, 359.$

Facts & Assumptions

Given: The factorisation $360 = 8 \cdot 9 \cdot 5$ into pairwise coprime prime powers.

[L1]

A unit is a square modulo $n$ if and only if it is a square modulo every prime-power factor of $n$ (A unit is a square modulo $n$ exactly when it is a square at every prime-power factor).

[L2]

The number of square roots of a soluble unit is the product of the local root counts (The number of square roots of a unit modulo $n$ is the product of the local counts).

Verification

technique · direct

1.1L1L2givenalgebra

The roots of $1$ are $1, 3, 5, 7$ modulo $8$ , are $1, 8$ modulo $9$ , and are $1, 4$ modulo $5$ . By [L1], every combination of these local roots gives a global root, and [L2] gives $4 \cdot 2 \cdot 2 = 16$ global roots.

2.1step 1.1L1L2algebra∎

Solving the finite CRT systems and reducing modulo $360$ gives exactly the displayed representatives. They are distinct, and reducing each one modulo $8$ , $9$ , and $5$ places it in the corresponding local root set from step 1.1, so [L1] verifies that each square is $1$ modulo $360$ ; the count in [L2] proves completeness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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 square roots of 11 modulo 360360 by the Chinese remainder theorem