# 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).
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## 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.