DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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

Definition

For $n \geq 1$ , Carmichael's function is the exponent of the finite unit group:

$λ (n) : = \exp ((Z / n)^{\times}) .$

The unit group is finite by The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ , and its exponent exists by The exponent of a finite group. In particular $λ (1) = 1$ , since the unit group modulo $1$ is trivial.

Depends on

Used by

If gcd(a,n)=1, then a^λ(n)≡1 pmod n Corollary
φ(8)=4 but every unit modulo 8 has square one Counterexample
Carmichael's λ(n) is the maximum order of a unit modulo n Proposition
Carmichael's function on prime powers and its least-common-multiple formula Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 14 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, Definition C.V.3 (standard reference, not scraped)

Carmichael's function λ(n)λ(n) as the exponent of (Z/nZ)×(Z/nZ)×