*D. H. Lehmer* [Bull. Am. Math. Soc. 38, 745-751 (1932;

Zbl 0005.34302)] raised the question whether or not the equation

$M\varphi \left(n\right)=n-1$ has any solutions if

$M>1$. Lehmer’s question has not yet been answered but some facts concerning solutions of the equation have been established. In the present paper the author improves some of the known facts. For example, he shows that if n is a solution and 3

$|n$, then n has at least 298 848 prime factors and

$n>{10}^{1937042}\xb7$ The presented proof is computer-aided. The author also proves that if, for a fixed value of M,

${S}_{Mt}$ is the set of solutions such that n has exactly t prime factors, then

${S}_{Mt}$ is finite (or empty).