Totient function

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In number theory, the totient function or Euler's φ function of a positive integer n, denoted φ(n), is defined to be the number of positive integers in the set {1,...,n} which are coprime to n. This function was studied by Leonhard Euler around 1730.[1]


The totient function is multiplicative and may be evaluated as


  • .
  • The average order of φ(n) is .

Euler's Theorem

The integers in the set {1,...,n} which are coprime to n represent the multiplicative group modulo n and hence the totient function of n is the order of (Z/n)*. By Lagrange's theorem, the multiplicative order of any element is a factor of φ(n): that is

  • if is coprime to .


  1. William Dunham, Euler, the Master of us all, MAA (1999) ISBN 0-8835-328-0. Pp.1-16.