Universal exponent

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 $n$ be a natural number. The universal exponent or Carmichael function of $n$, denoted $\lambda (n)$ is defined in the following equivalent ways:

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

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

Relation with other arithmetic functions

• Euler phi-function: The universal exponent $\lambda (n)$ divides the Euler phi-function $\phi (n)$. 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.