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.