On the Nagell-Ljunggren equation $(x^n - 1)/(x - 1) = y^q$

Abstract

We establish several new results on the Nagell-Ljunggren equation $(x^n - 1)/(x-1) = y^q$. Among others, we prove that, for every solution $(x, y, n, q)$ to this equation, $n$ has at most four prime divisors, counted with their multiplicities.