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 $(p^n)^x + (3^mp + 2)^y = z^2$
Résumé
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 $(x, y, z) = (5, 4, 122)$.Téléchargements
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Références
\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).
Publiée
2026-02-23
Rubrique
Conf. Issue: Advances in Algebra, Analysis, Optimization, and Modeling
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