ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How 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 positive moduli below $20$ admitting primitive roots are $1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19$

Example

The positive integers below $20$ that admit primitive roots are

$1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19.$

Facts & Assumptions

Given: The positive integers $n < 20$ .

[L1]

A modulus admits a primitive root exactly when it is $1$ , $2$ , $4$ , an odd prime power, or twice an odd prime power (A positive integer admits a primitive root exactly when it is $1$ , $2$ , $4$ , $p^{k}$ , or $2 p^{k}$ for an odd prime $p$ ).

Verification

technique · direct

1.1L1algebra

Below $20$ , the odd prime powers are $3, 5, 7, 9, 11, 13, 17, 19$ , and twice such a power gives $6, 10, 14, 18$ ; together with $1, 2, 4$ this is the displayed list.

2.1step 1.1L1algebra∎

The omitted positive integers are $8, 12, 15, 16$ : $8$ and $16$ are powers $2^{a}$ with $a \geq 3$ , $12 = 4 \cdot 3$ , and $15$ has two odd prime factors, so [L1] excludes each.

Depends on

A positive integer admits a primitive root exactly when it is $1$, $2$, $4$, $p^k$, or $2p^k$ for an odd prime $p$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 positive moduli below 2020 admitting primitive roots are 1,2,3,4,5,6,7,9,10,11,13,14,17,18,191,2,3,4,5,6,7,9,10,11,13,14,17,18,19