CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Every prime modulus admits a primitive root

Statement

Every prime $p$ admits a primitive root modulo $p$ .

Facts & Assumptions

Given: A prime $p$ .

[L1]

The unit group $(Z / p)^{\times}$ is a finite cyclic group (For every prime $p$ , the multiplicative group $(Z / p Z)^{\times}$ is cyclic).

[L2]

A unit is a primitive root exactly when it generates the unit group (A unit is a primitive root modulo $n$ if and only if it generates $(Z / n Z)^{\times}$ ).

Proof

technique · direct

1.1

Choose a generator $g$ of the cyclic group in [L1]; such a generator exists also when $p = 2$ , since the one-element group is cyclic.

L1choose

2.1

By [L2], the chosen $g$ is a primitive root modulo $p$ .

step 1.1L2∎

Depends on

Used by

For every odd prime p and k≥1, (ℤ/pᵏℤ)^× is cyclic of order pᵏ⁻¹(p-1) Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 11 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.

Sources

Peter Hackman, Elementary Number Theory, Theorem C.II.1 (standard reference, not scraped)
William Stein, Elementary Number Theory, Theorem 2.5.8 (standard reference, not scraped)