DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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 (mod 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 · two levels

12 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

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)×