# Totient function

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Euler totient function, Euler totient

Another frequently used named for the Euler function , which counts the natural numbers that are relatively prime to .

The Carmichael conjecture on the Euler totient function states that if , then for some ; i.e. no value of the Euler function is assumed once. This has now been verified for , [a1].

A natural generalization of the Euler totient function is the Jordan totient function , which counts the number of -tuples , , such that . Clearly, .

One has where runs over the prime numbers dividing , and where is the Möbius function and runs over all divisors of . For these formulas reduce to the well-known formulas for the Euler function.

The Lehmer problem on the Euler totient function asks for the solutions of , , [a2]. For some results on this still (1996) largely open problem, see [a3] and the references therein. The corresponding problem for the Jordan totient function (and ) is easy, [a4]: For , if and only if is a prime number. Moreover, if is a prime number, then .

For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [a5], [a6].

How to Cite This Entry:
Totient function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totient_function&oldid=12673
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article