Alphabeta MathHow 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.
The square roots of modulo by the Chinese remainder theorem
Example
The square roots of modulo are
Facts & Assumptions
Given: The factorisation into pairwise coprime prime powers.
A unit is a square modulo if and only if it is a square modulo every prime-power factor of (A unit is a square modulo exactly when it is a square at every prime-power factor).
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 is the product of the local counts).
Verification
The roots of are modulo , are modulo , and are modulo . By [L1], every combination of these local roots gives a global root, and [L2] gives global roots.
Solving the finite CRT systems and reducing modulo gives exactly the displayed representatives. They are distinct, and reducing each one modulo , , and places it in the corresponding local root set from step 1.1, so [L1] verifies that each square is modulo ; the count in [L2] proves completeness.
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: 48 results over 13 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.