Totient function
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].
References
[a1] | A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below ![]() |
[a2] | D.H. Lehmer, "On Euler's totient function" Bull. Amer. Math. Soc. , 38 (1932) pp. 745–751 |
[a3] | V. Siva Rama Prasad, M. Rangamma, "On composite ![]() ![]() |
[a4] | M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" Rocky Mount. J. Math. , 15 (1985) pp. 609–620 |
[a5] | R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188 |
[a6] | R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190 |
[a7] | L.E. Dickson, "History of the theory of numbers" , I: Divisibility and primality , Chelsea, reprint (1971) pp. Chapt. V; 113–155 |
Totient function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totient_function&oldid=12673