Akademik Veri Yönetim Sistemi
International Conference on Diophantine equations, Polynomials and Related Areas, New Delhi, Hindistan, 25 - 29 Kasım 2024, ss.1
Let $d$ and $\delta$ be fixed positive integers. Consider the Diophantine equation $x^2+d^s=\delta y^n$ where $x, y, n$ and $s$ are nonnegative integer unknowns. This equation is usually called the generalized Lebesgue-Ramanujan-Nagell equation. It has a long history and rich content. Recently, a survey paper on the generalized Lebesgue-Ramanujan-Nagell equation has been written by M.-H. Le and G. Soydan, \cite{LS}.
Denote by $h=h(-p)$ the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ with $p$ prime. It is well known that $h=1$ for $p\in\{3,7,11,19,43,67,163\}$. Recently, all the solutions of the Diophantine equation $x^2+p^s=4y^n$ with $h=1$ were given by Chakraborty et al. in \cite{Cetal}. In this talk, we consider the Diophantine equation $x^2+p^s=2^ry^n$ in unknown integers $(x,y,s,r,n)$ where $s\ge 0$, $r\geq 3$, $n \geq 3$, $h\in\{1,2,3\}$ and $\gcd(x,y)=1$. Our main tools include the known results from the modularity of Galois representations associated with Frey-Hellegoaurch elliptic curves (i.e. modular approach) \cite{BS}, the symplectic method \cite{FK}, a Thue-Mahler solver which was improved by Gherga and Siksek \cite{GheSi} in $\mathtt{MAGMA}$ \cite{Magma} and elementary methods of classical algebraic number theory. This work was supported by the Research Fund of Bursa Uluda\u{g} University under Project No: FGA-2023-1545. This is a joint work with Elif K{\i}z{\i}ldere Mutlu, \cite{MS}.
\bibitem{BS} {\sc M. A. Bennett and C. Skinner}, Ternary Diophantine equations via Galois representations and modular forms, {\it Canad.~J.~Math.} {\bf 56} (2004) 23--54.
\bibitem{Magma} {\sc W. Bosma, J. Cannon, C. Playoust}, The Magma Algebra System I. The user language, {\em J. Symbolic Comput.} {\bf 24} (1997), 235--265.
\bibitem{Cetal} {\sc K. Chakraborty, A. Hoque and R. Sharma}, Complete Solutions of Certain Lebesgue-Ramanujan-Nagell Type equations, {\it Publ. Math. Debrecen}, {\bf 97}(3-4) (2020), 339--352.
\bibitem{FK} N. Freitas, A. Kraus, On the symplectic type of isomorphism of the p-torsion of elliptic curves, {\it Mem. Amer. Math. Soc.} {\bf 277} (2022) 1--104.
\bibitem{GheSi} {\sc A. Gherga and S. Siksek}, Efficient resolution of Thue-Mahler equations, {\it arXiv preprint} {https://arxiv.org/pdf/arXiv:2207.14492}{arXiv:2207.14492} 2022.
\bibitem{LS} {\sc M. H. Le and G. Soydan}, A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation, {\em Surv. Math. Appl.}, {\bf 15} (2020), 473--523.
\bibitem{MS} {\sc E.K. Mutlu and G. Soydan}, On the solution of some generalized Lebesgue-Ramanujan-Nagell type equations, {\em Int. Journal of Number Theory}, {\bf 20} (2024), 1195--1218.