On the Diophantine Equation (p + 4n)^ x + p^y = z^2

Authors

  • Wachirarak Orosram Department of Mathematics Faculty of Science Buriram Rajabhat University
  • Kitsanuphong Makonwattana
  • Saichon Khongsawat

DOI:

https://doi.org/10.29020/nybg.ejpam.v15i4.4508

Keywords:

exponential Diophantine equation, Catalan’s conjecture

Abstract

In this paper, we study the Diophantine equation $(p+4n)^x+p^y=z^2,$ where $n$ is a non-negative integer and $p, p+4n$ are prime numbers such that $p\equiv 7\pmod{12}$. We show that the non-negative integer solutions of such equation are $(x, y, z)\in
\{(0, 1, \sqrt {p+1})\} \cup \{ (1, 0, 2\sqrt{n+\frac{p+1}{4}})\}$, where $\sqrt {p+1}$ and $\sqrt{n+\frac{p+1}{4}}$ are integers.

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How to Cite

Orosram, W., Makonwattana, K., & Khongsawat, S. (2022). On the Diophantine Equation (p + 4n)^ x + p^y = z^2. European Journal of Pure and Applied Mathematics, 15(4), 1593–1596. https://doi.org/10.29020/nybg.ejpam.v15i4.4508