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The diophantine equation $\left(2^{k}-1\right)\left(3^{k}-1\right)=x^{n}$

Number Theory 2025-07-29 v2

Abstract

In this paper, we investigate the Diophantine equation (2k1)(3k1)=xn (2^k - 1)(3^k - 1) = x^n and prove that it has no solutions in positive integers k,x,n>2k, x, n > 2.

Keywords

Cite

@article{arxiv.2501.04050,
  title  = {The diophantine equation $\left(2^{k}-1\right)\left(3^{k}-1\right)=x^{n}$},
  author = {Bo He and Chang Liu},
  journal= {arXiv preprint arXiv:2501.04050},
  year   = {2025}
}

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6 pages