On the Diophantine equation $(p^n)^x + (3^mp + 2)^y = z^2,$ where $p$ and $3^mp + 2$ are prime numbers
The Diophantine equation $(p^n)^x + (3^mp + 2)^y = z^2$
Abstract
The main objective of this paper aims to give methods for find the Diophantine equation $(p^n)^x + (3^mp + 2)^y = z^2$ in $\mathbb{N}$, where the parameters $p \geq 3$ and $3^mp + 2$ are prime integers. Concretely, we employ a congruence method, and we investigate that the nonexistence solutions of this equation for a prime $p > 3$. Subsequently, we will establish that this equation has no solutions for the prime $p = 3$ and any $m > 1$. In the sequel, for $m = 1$, an analysis via the elliptic curves reveals that if $n = 1$, this equation has a unique solution, given by $(x, y, z) = (5, 4, 122)$.Downloads
Download data is not yet available.
References
\bibitem{bib1} S. A. Arif and F. S. Abu Muriefah,
{\it The diophantine equation $x^2 + 3^m = y^n$}. Internat. J. Math. Sci. {\bf 21}(3), 619--620, (1968).
\bibitem{bib2} S. A. Arif and F. S. Abu Muriefah,
{\it On the Diophantine equation $x^2 + q^{2k + 1} = y^n$}. J. Number Theory {\bf 95}(1), 95--100, (2002).
\bibitem{bib3} E. Catalan,
{\it A note on extraite dune lettre adressee a lediteur}. J. Reine Angew. Math. {\bf 27}, 192, (1844).
\bibitem{bib4} J. H. E. Cohn,
{\it The diophantine equation $x^2 + 2^k = y^n$}. Arch. Math. {\bf 59}(4), 341--344, (1992).
\bibitem{bib5} M.-J. Deng,
{\it A note on the Diophantine equation $x^2 + q^m = c^{2n}$}. Proc. Japan Acad. Ser. A Math. Sci. {\bf 91}(2), 15--18, (2015).
\bibitem{bib6} F. Luca,
{\it On a diophantine equation}. Bull. Austral. Math. Soc. {\bf 61}(2), 241--246, (2000).
\bibitem{bib7} P. Mih\u{a}ilescu,
{\it On primary Cyclotomic units and a proof of Catalan's conjecture}. J. Reine Angew. Math. {\bf 572}, 167--195, (2004).
\bibitem{bib9} P. H. Nam,
{\it On the Diophantine equation $(p^n)^x + (4^mp + 1)^y = z^2$ when $p$, $4^mp + 1$ are prime numbers}. Int. Electron. J. Algebra {\bf 37}(37), 190--200, (2025).
\bibitem{bib10} W. A. Stein,
{\em Sage Mathematics Software (Version 10.4)}. The Sage Development Team (2023). http://www.sagemath.org/
\bibitem{bib12} N. Terai,
{\it The Diophantine equation $x^2 + q^m = p^n$}. Acta Arith. {\bf 63}(4), 351--358, (1993).
\bibitem{bib13} N. Terai,
{\it A note on the Diophantine equation $x^2 + q^m = c^n$}. Bull. Aust. Math. Soc. {\bf 90}(1), 20--27, (2014).
\bibitem{bib14} H. Zhu,
{\it A note on the Diophantine equation $x^2 + q^m = y^3$}. Acta Arith. {\bf 146}(2), 195--202, (2011).
{\it The diophantine equation $x^2 + 3^m = y^n$}. Internat. J. Math. Sci. {\bf 21}(3), 619--620, (1968).
\bibitem{bib2} S. A. Arif and F. S. Abu Muriefah,
{\it On the Diophantine equation $x^2 + q^{2k + 1} = y^n$}. J. Number Theory {\bf 95}(1), 95--100, (2002).
\bibitem{bib3} E. Catalan,
{\it A note on extraite dune lettre adressee a lediteur}. J. Reine Angew. Math. {\bf 27}, 192, (1844).
\bibitem{bib4} J. H. E. Cohn,
{\it The diophantine equation $x^2 + 2^k = y^n$}. Arch. Math. {\bf 59}(4), 341--344, (1992).
\bibitem{bib5} M.-J. Deng,
{\it A note on the Diophantine equation $x^2 + q^m = c^{2n}$}. Proc. Japan Acad. Ser. A Math. Sci. {\bf 91}(2), 15--18, (2015).
\bibitem{bib6} F. Luca,
{\it On a diophantine equation}. Bull. Austral. Math. Soc. {\bf 61}(2), 241--246, (2000).
\bibitem{bib7} P. Mih\u{a}ilescu,
{\it On primary Cyclotomic units and a proof of Catalan's conjecture}. J. Reine Angew. Math. {\bf 572}, 167--195, (2004).
\bibitem{bib9} P. H. Nam,
{\it On the Diophantine equation $(p^n)^x + (4^mp + 1)^y = z^2$ when $p$, $4^mp + 1$ are prime numbers}. Int. Electron. J. Algebra {\bf 37}(37), 190--200, (2025).
\bibitem{bib10} W. A. Stein,
{\em Sage Mathematics Software (Version 10.4)}. The Sage Development Team (2023). http://www.sagemath.org/
\bibitem{bib12} N. Terai,
{\it The Diophantine equation $x^2 + q^m = p^n$}. Acta Arith. {\bf 63}(4), 351--358, (1993).
\bibitem{bib13} N. Terai,
{\it A note on the Diophantine equation $x^2 + q^m = c^n$}. Bull. Aust. Math. Soc. {\bf 90}(1), 20--27, (2014).
\bibitem{bib14} H. Zhu,
{\it A note on the Diophantine equation $x^2 + q^m = y^3$}. Acta Arith. {\bf 146}(2), 195--202, (2011).
Published
2026-02-23
Section
Conf. Issue: Advances in Algebra, Analysis, Optimization, and Modeling
Copyright (c) 2026 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
