Lehmer's Totient Problem
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).
REFERENCES:
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.
