The Nagell–Ljunggren equation via Runge’s method

@article{Bennett2013TheNE,
  title={The Nagell–Ljunggren equation via Runge’s method},
  author={Michael A. Bennett and Aaron Levin},
  journal={Monatshefte f{\"u}r Mathematik},
  year={2013},
  volume={177},
  pages={15-31}
}
The Diophantine equation $$\frac{x^n-1}{x-1}=y^q$$xn-1x-1=yq has four known solutions in integers $$x, y, q$$x,y,q and $$n$$n with $$|x|, |y|, q > 1$$|x|,|y|,q>1 and $$n > 2$$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 $$(x,y,n,q)$$(x,y,n,q) is a solution to this equation, then $$n$$n has three or fewer prime divisors, counted with multiplicity. This improves a result of Bugeaud and Mihăilescu… 
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