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