Lehmer's totient problem

In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem.

It is known that φ(n) = n − 1 if and only if n is prime. So for every prime number n, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1. D. H. Lehmer conjectured in 1932 that there are no composite numbers with this property.

History

 * Lehmer showed that if any composite solution n exists, it must be odd, square-free, and divisible by at least seven distinct primes (i.e. ω(n) ≥ 7). Such a number must also be a Carmichael number.
 * In 1980, Cohen and Hagis proved that, for any solution n to the problem, n > 1020 and ω(n) ≥  14.
 * In 1988, Hagis showed that if 3 divides any solution n, then n > 10$1,937,042$ and ω(n) ≥ $298,848$. This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution n, then n > 10$360,000,000$ and ω(n) ≥ $40,000,000$.
 * A result from 2011 states that the number of solutions to the problem less than $$X$$ is at most $${X^{1/2}/(\log X)^{1/2+o(1)}}$$.