Lehmer's totient problem

In mathematics, Lehmer's totient problem asks whether there is any composite number such that Euler's totient function divides. This is an unsolved problem.
It is known that if and only if is prime. So for every prime number, we have and thus in particular divides. D. H. Lehmer asked in 1932 whether there exist composite numbers with this property.

History

Lehmer showed that if any composite solution ' exists, it must be odd, square-free, and divisible by at least seven distinct primes. Such a number must also be a Carmichael number.
In 1980, Cohen and Hagis proved that, for any solution ' to the problem, and.
In 1988, Hagis showed that if 3 divides any solution ', then and. This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution ', then and.
A result from 2011 states that the number of solutions to the problem less than is at most.