CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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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If $\gcd (a, n) = 1$ , then $a^{λ (n)} \equiv 1 (m o d n)$

Statement

If $n \geq 1$ and $\gcd (a, n) = 1$ , then

$a^{λ (n)} \equiv 1 (m o d n) .$

Facts & Assumptions

Given: Integers $n \geq 1$ and $a$ with $\gcd (a, n) = 1$ .

[L1]

$λ (n)$ is the exponent of $(Z / n)^{\times}$ (Carmichael's function $λ (n)$ as the exponent of $(Z / n Z)^{\times}$ ).

[L2]

The class of $a$ is a unit exactly when $\gcd (a, n) = 1$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ ).

[L3]

Every element of a finite group raised to its exponent is the identity (The exponent of a finite group).

Proof

technique · direct

1.1

By [L2], the class of $a$ lies in the unit group modulo $n$ .

2.1

By [L1] and [L3], its $λ (n)$ th power is the identity class, which is the asserted congruence. For $n = 1$ , both sides are the unique class and $λ (1) = 1$ .

step 1.1L1L3∎

Depends on

Used by

λ(561)=80 and every integer coprime to 561 has eightieth power congruent to one Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 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.

Sources

Peter Hackman, Elementary Number Theory, §C.V (standard reference, not scraped)

If gcd⁡(a,n)=1gcd(a,n)=1, then aλ(n)≡1(modn)aλ(n)≡1(modn)