The Nagell-Ljunggren equation via Runge's method

TL;DR

This paper investigates the solutions of the Nagell-Ljunggren equation, proving that any solution must have n with at most three prime factors, thus narrowing the possible solutions and advancing understanding of this Diophantine problem.

Contribution

The paper improves previous results by showing that solutions to the equation have n with at most three prime divisors, using Runge's method.

Findings

Any solution has n with at most three prime divisors

The result narrows the search for solutions to the equation

Advances the understanding of the structure of solutions

Abstract

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.