The Nagell–Ljunggren equation via Runge’s method

  title={The Nagell–Ljunggren equation via Runge’s method},
  author={M. A. Bennett and A. Levin},
The Diophantine equation x n−1 x−1 = yq 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 Mihăilescu. 

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