Lehmer's totient problem asks if there exist any composite numbers
such that ,
where
is the totient function? No such numbers are
known. However, any such an would need to be a Carmichael
number, since for every element in the integers (mod ), , so and is a Carmichael number.

In 1932, Lehmer showed that such an must be odd and squarefree,
and that the number of distinct prime factors
must satisfy .
This was subsequently extended to . The best current result is and , improving the lower bound of Cohen and Hagis (1980) since there are
no Carmichael numbers less than having distinct prime
factors; Pinch). However, even better results are known in the special cases
,
in which case
(Wall 1980), and , in which case and (Lieuwens 1970).

Cohen, G. L. and Hagis, P. Jr. "On the Number of Prime Factors of is ." Nieuw Arch. Wisk.28, 177-185,
1980.Cohen, G. L. and Segal, S. L. "A Note Concerning
Those
for which
Divides ."
Fib. Quart.27, 285-286, 1989.Lieuwens, E. "Do There
Exist Composite Numbers for Which Holds?" Nieuw Arch. Wisk.18,
165-169, 1970.Pinch, R. G. E. ftp://ftp.dpmms.cam.ac.uk/pub/Carmichael/table.Ribenboim,
P. The
New Book of Prime Number Records. New York: Springer-Verlag, pp. 27-28,
1989.Wall, D. W. "Conditions for to Properly Divide ." In A
Collection of Manuscripts Related to the Fibonacci Sequence (Ed. V. E.
Hoggatt and M. V. E. Bicknell-Johnson). San Jose, CA: Fibonacci Assoc.,
pp. 205-208, 1980.