# Universal exponent

From Number

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).

View a complete list of arithmetic functions

## Definition

Let be a natural number. The **universal exponent** or **Carmichael function** of , denoted is defined in the following equivalent ways:

- It is the exponent of the multiplicative group modulo .
- It is the least common multiple of the orders, modulo , of all integers relatively prime to .
- It is the largest possible order, modulo , of an integer relatively prime to .

The symbol is also used for the Liouville lambda-function, which is totally different, while the capital letter is used for the von Mangoldt function, which is totally different too.

## Relation with other arithmetic functions

- Euler phi-function: The universal exponent divides the Euler phi-function . This can be thought of as a reformulation of Euler's theorem, and is the group-theoretic fact that the exponent of a group divides the order of the group.