Solution of the Diophantine equation $x^2 + p^k=y^n$

Arkabrata Ghosh

Solution of the Diophantine equation $x^2 + p^k=y^n$

Number Theory 2024-03-25 v2

Authors: Arkabrata Ghosh

Abstract

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$ , $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$ . In this article, I used a method involving prime factorization and class numbers which is different from using congruent number argument which is widely used in this type of problem.

Keywords

diophantine equation class number prime factorization imaginary quadratic field perfect square number theory congruent number

Cite

@article{arxiv.2402.19445,
  title  = {Solution of the Diophantine equation $x^2 + p^k=y^n$},
  author = {Arkabrata Ghosh},
  journal= {arXiv preprint arXiv:2402.19445},
  year   = {2024}
}

Comments

Results of this article is a particular case of some other paper already published. I did not knew at the time of uploading this article. So there is no new contribution made in this paper. That is why I want to remove it

Solution of the Diophantine equation $x^2 + p^k=y^n$

Abstract

Keywords

Cite

Comments

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