# Carmichael Function λ(n) Calculator

Group exponent λ(n) from prime factorization with φ(n) comparison, primitive-root existence, Korselt's Carmichael-number detection, and optional sample-unit verification.

> Canonical page: https://elysiatools.com/en/tools/carmichael-function

- **Category:** Math & Numbers

- **Keywords:** carmichael function, lambda function, exponent of group, multiplicative group, carmichael number, korselt criterion, primitive root, number theory

## Overview

The Carmichael Function λ(n) Calculator determines the universal exponent of the multiplicative group modulo n from its prime factorization. It computes per-prime-power exponents, compares the resulting group exponent λ(n) against Euler's totient φ(n), determines whether a primitive root exists, checks Korselt's criterion for Carmichael numbers, and optionally verifies modular congruences against sample coprime units.

## Inputs

- **Number n** (text): Positive integer, 1 ≤ n ≤ 10¹² (trial-division factorization bound).
- **Output detail** (select)

## When to use

- Finding the minimal universal exponent λ(n) such that a^λ(n) ≡ 1 (mod n) for all integers coprime to n.
- Checking whether a composite modulus is a Carmichael number via Korselt's criterion (λ(n) divides n − 1).
- Determining if a given modulus n possesses a primitive root by comparing λ(n) with Euler's totient φ(n).

## How it works

- Decomposes the input integer n into its prime power factorization p^k.
- Calculates component exponents using λ(2) = 1, λ(4) = 2, λ(2^k) = 2^(k−2) for k ≥ 3, and λ(p^k) = φ(p^k) for odd primes.
- Takes the least common multiple (lcm) of all component exponents to determine the overall group exponent λ(n).
- Compares λ(n) to φ(n), evaluates Korselt's criterion, and optionally verifies the congruence a^λ(n) ≡ 1 (mod n) with sample coprime bases.

## Use cases

- Optimizing cryptographic private exponents and analyzing modular arithmetic cycles in RSA architectures.
- Validating abstract algebra coursework involving multiplicative group structures, subgroup orders, and primitive roots.
- Analyzing Fermat pseudoprimes and verifying Carmichael number candidates in number theory research.

## Frequently asked questions

### What is the difference between Carmichael's λ(n) and Euler's φ(n)?

Euler's φ(n) is the order of the multiplicative group modulo n, while Carmichael's λ(n) is the smallest positive exponent satisfying a^λ(n) ≡ 1 (mod n) for all coprime integers a. λ(n) always divides φ(n) and is often smaller.

### When does an integer n have a primitive root?

A primitive root exists if and only if λ(n) = φ(n), which occurs solely when n is 1, 2, 4, p^k, or 2p^k for an odd prime p.

### How does the calculator identify a Carmichael number?

It tests Korselt's criterion: n must be composite, square-free, and satisfy the condition that λ(n) divides n − 1.

### What is the maximum supported input value for n?

The calculator supports positive integers n up to 10¹² using trial-division prime factorization.

### What does the sample-unit verification option do?

It generates coprime units modulo n and explicitly computes a^λ(n) mod n to verify that each result equals 1.

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