Abstract
Let k be a fixed positive integer with \(k>1\). In this paper, using various elementary methods in number theory, we give criteria under which the equation \(x^2+(2k-1)^y=k^z\) has no positive integer solutions (x, y, z) with \(y\in \{3,5\}\).
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References
M. Bennett and N. Billery, Sums of two S-units via Frey-Hellegouarch curves, Math. Comp. 305 (2017), 1375–1401.
B. Çelik, Maple ve Maple ile Matematik, Dora Yayın Dağıtım, Bursa, 2014.
M.-J. Deng, J. Guo and A.-J. Xu, A note on the Diophantine equation \(x^2+(2c-1)^m=c^n\), Bull. Aust. Math. Soc. 98 (2018), 188–195.
Y. Fujita and M. Le, On a conjecture concerning the generalized Ramanujan-Nagell equation, J. Comb. Number Theory 12 (2020), 1–10.
Y. Fujita and N. Terai, A note on the Diophantine equation \(x^2+q^m=c^n\), Acta Math. Hung. 162 (2020), 518–526.
M.-H. Le, Some exponential Diophantine equation I: The equation \(D_1x^2-D_2y^2=\lambda k^z\). J. Number Theory 55 (1995), 209–221.
M. H. Le and G. Soydan, A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation, Surv. Math. Appl. 15 (2020), 473–523.
E. K. Mutlu, M. H. Le and G. Soydan, A modular approach to the generalized Lebesgue-Ramanujan-Nagell equation, Indagationes Mathematicae 33 (2022), 992–1000.
N. Terai, A note on the Diophantine equation \(x^2+q^m=c^n\), Bull. Aust. Math. Soc. 90 (2014), 20–27.
H. Yang and R.-Q. Fu, An upper bound for least solutions of exponential Diophantine equation \(D_1x^2-D_2y^2=\lambda k^z\), Int. J. Number Theory 11 (2015), 1107–1114.
Acknowledgements
We would like to thank Professor Nikos Tzanakis for useful discussions and anonymous referee for carefully reading our paper and for his/her corrections. The first author is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) 2211/A National PhD scholarship program.
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Communicated by B. Sury.
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Mutlu, E.K., Le, M. & Soydan, G. An elementary approach to the generalized Ramanujan–Nagell equation. Indian J Pure Appl Math 55, 392–399 (2024). https://doi.org/10.1007/s13226-023-00372-8
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DOI: https://doi.org/10.1007/s13226-023-00372-8
Keywords
- Polynomial-exponential Diophantine equation
- Generalized Ramanujan–Nagell equation
- Elementary method in number theory