CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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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.

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A modulus with primitive roots has exactly $φ (φ (n))$ primitive roots

Statement

If $n \geq 1$ admits a primitive root, then it has exactly $φ (φ (n))$ primitive roots.

Facts & Assumptions

Given: A positive modulus $n$ admitting a primitive root.

[L1]

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

[L2]

A cyclic group of order $m$ has $φ (m)$ generators (The generators of a cyclic group of order $m$ are the $g^{a}$ with $\gcd (a, m) = 1$ , so there are $φ (m)$ of them).

Proof

technique · direct

1.1givenL1

By the Given and [L1], the unit group is cyclic of order $φ (n)$ , and its generators are exactly the primitive roots.

2.1step 1.1L2∎

Applying [L2] with $m = φ (n)$ yields $φ (φ (n))$ primitive roots. At $n = 1$ this is $φ (1) = 1$ , counting the unique class.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.

Sources

William Stein, Elementary Number Theory, Proposition 2.5.12 (standard reference, not scraped)

A modulus with primitive roots has exactly φ(φ(n))φ(φ(n)) primitive roots