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
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 .
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.