Perfect powers in sum of three fifth powers

Pranabesh Das; Pallab Kanti Dey; Angelos Koutsianas; Nikos Tzanakis

Perfect powers in sum of three fifth powers

Number Theory 2020-08-31 v2

Authors: Pranabesh Das , Pallab Kanti Dey , Angelos Koutsianas , Nikos Tzanakis

Abstract

In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $(x-d)^5 + x^5 + (x + d)^5 = z^n,~n\geq 2,$ where $d,x,z \in \mathbb{Z}$ and $d = 2^a5^b$ with $a,b\geq 0$ .

Keywords

perfect powers fifth powers diophantine equation arithmetic progression sums of powers

Cite

@article{arxiv.2008.07804,
  title  = {Perfect powers in sum of three fifth powers},
  author = {Pranabesh Das and Pallab Kanti Dey and Angelos Koutsianas and Nikos Tzanakis},
  journal= {arXiv preprint arXiv:2008.07804},
  year   = {2020}
}

Comments

21 pages, minor modifications in version 2

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