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).
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. 
to Properly Divide 
." In