  The Diophantine equation (x^n-1)/(x-1)=y^q has four known solutions in
integers x, y, q and n with |x|, |y|, q > 1 and n > 2. Whilst we expect that
there are, in fact, no more solutions, such a result is well beyond current
technology. In this paper, we prove that if (x,y,n,q) is a solution to this
equation, then n has three or fewer prime divisors, counted with multiplicity.
This improves a result of Bugeaud and Mihailescu.
