Carmichael Function λ(n) Calculator

Carmichael Function λ(n) Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text, select
Output type: text
Sample coverage: 4
API ready: Yes

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.

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

1Decomposes the input integer n into its prime power factorization p^k.
2Calculates 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.
3Takes the least common multiple (lcm) of all component exponents to determine the overall group exponent λ(n).
4Compares 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.

Examples

1. Evaluating the Smallest Carmichael Number

Number Theory Student

Background

Studying Fermat pseudoprimes and Korselt's criterion in an abstract algebra course.

Problem

Needs to verify why 561 passes Fermat's test for all coprime bases despite being composite.

How to use

Enter 561 for Number n and select the verification detail option.

n = 561, detail = verify

Outcome

Computes prime factors 3 × 11 × 17, finds λ(561) = lcm(2, 10, 16) = 80, confirms λ(561) | 560, and verifies sample modular congruences.

2. Analyzing Multiplicative Group Modulo 8

Cryptography Researcher

Background

Investigating exponent collapse in powers of two for modular arithmetic optimizations.

Problem

Determine the group exponent for (Z/8Z)* to demonstrate why odd squares are always 1 modulo 8.

How to use

Enter 8 for Number n and keep the standard factorization detail option.

FAQ

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.

n = 8, detail = value

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Evaluating the Smallest Carmichael Number

2. Analyzing Multiplicative Group Modulo 8

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