The Nagell-Ljunggren equation via Runge's method
Number Theory
2013-12-17 v1
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.
Keywords
Cite
@article{arxiv.1312.4037,
title = {The Nagell-Ljunggren equation via Runge's method},
author = {Michael A. Bennett and Aaron Levin},
journal= {arXiv preprint arXiv:1312.4037},
year = {2013}
}