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.
Odd-prime tests alone miss the obstruction to being a square modulo
Statement refuted
For an even modulus, testing whether a unit is a square modulo every odd prime divisor does not suffice to prove that it is a square modulo the original modulus.
Facts & Assumptions
Given: The unit modulo .
For , an odd integer is a square modulo if and only if it is congruent to modulo (Unit square criterion and root count modulo powers of two).
A unit square modulo 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).
Counterexample
The only odd prime divisor of is , and , so it passes the odd-prime square test. But , and [L1] shows that it is not a square modulo .
If were a square modulo , the forward implication in [L2] would make it a square modulo the prime-power factor , contradicting step 1.1. Hence the odd-prime test misses the two-part obstruction.
Depends on
Used by
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