# CarmichaelLambda

gives the Carmichael function .

# Details • CarmichaelLambda is also known as the reduced totient function or the least universal exponent function.
• CarmichaelLambda is typically used in primality testing to find a composite number that cannot be proved composite by some primality tests.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• is the smallest positive integer such that for all relatively prime to .
• For a number with a unit and primes, returns LCM[(p1-1) ,,(pm-1) ].
• # Examples

open allclose all

## Basic Examples(2)

Compute CarmichaelLambda of :

Plot the sequence:

## Scope(7)

### Numerical Manipulation(4)

Compute using integers:

Compute for large integers:

### Symbolic Manipulation(3)

Find an integer solution instance:

Simplify an expression:

Identify the CarmichaelLambda sequence:

## Applications(7)

### Basic Applications(3)

The first 20 values of CarmichaelLambda:

Discrete plot:

Number line plot:

Plot the generating function:

Exponential generating function:

Dirichlet series:

### Primality Testing(2)

Given a prime, for all positive numbers a less than p:

A natural test for primality:

This test can be inconclusive for composite integers n satisfying :

Verify for all a coprime to 561:

Recognize Carmichael numbers, composite numbers with an1 mod n for all a coprime to n:

The number is a Carmichael number, is not:

### Cryptography(1)

Find the universal exponent of the multiplication group modulo n:

Private key:

Public key:

Encrypt a message:

Decrypt it:

### Number Theory(1)

Find the number of elements in the largest subgroup of Z_n^*:

## Properties & Relations(7)

The result is non-negative:

Divisibility is preserved:

The LCM of CarmichaelLambda is equal to CarmichaelLambda of the LCM:

If is square-free then aaλ(n)+1mod n:

The multiplicative order of an element modulo divides :

CarmichaelLambda divides EulerPhi:

If has a primitive root, then CarmichaelLambda and EulerPhi are the same:

## Neat Examples(2)

A plot of varying CarmichaelLambda values:

Ulam spiral where numbers are colored based on the values of CarmichaelLambda: