Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

Murat Alan; Mustafa Aydın

doi:10.33434/cams.1675491

EN

Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

Abstract

In this paper, we provide a complete classification of the integer solutions to the Diophantine equation $x^2 + 3^a 19^b 73^c = \lambda y^n,$ where $ \lambda = 2^{\delta} $, $ x, y \geq 1 $, $ a, b, c, \delta \geq 0 $, $ n \geq 3 $, $ \gcd(x, y) = 1 $. Our approach combines the Primitive Divisor Theorem for Lehmer sequences, proved by Bilu, Hanrot, and Voutier, with fundamental properties of algebraic integer rings. By employing these methods, we determine all possible solutions in non-negative integers.

Keywords

Diophantine equations, Elliptic curves, Lehmer sequences, Primitive divisor theorem, Quartic curves, Ramanujan-Nagell equations

References

[1] M. Le, G. Soydan, A brief survey on the generalized Lebesgue–Ramanujan–Nagell equation, Surv. Math. Appl., 15 (2020), 473--523.
[2] M. Alan, U. Zengin, On the Diophantine equation $x^2+3^a41^b=y^n$, Period. Math. Hungar., 81 (2020), 284--291. https://doi.org/10.1007/s10998-020-00321-6.
[3] M. Alan, M. Aydın, On the Diophantine equation $x^2 + 2^a 3^b 73^c= y^n$, Arch. Math., 59(5) (2023), 411--420. https://doi.org/10.5817/AM2023-5-411.
[4] S. A. Arif, F. S. Abu Muriefah, On the Diophantine equation $x^2+q^{2k+1}=y^n$, J. Number Theory, 95(1) (2002), 95--100. https://doi.org/10.1006/jnth.2001.2750.
[5] A. Bérczes, I. Pink, On the Diophantine equation $x^2 + d^{2\ell+1} = y^n$, Glasgow Math. J., 54 (2012), 415--428. https://doi.org/10.1017/S0017089512000067.
[6] I. Cangül, M. Demirci, I. Inam, F. Luca, G. Soydan, On the Diophantine equation $x^2+ 2^a 3^b 11^c= y^n$, Math. Slovaca, 63(3) (2013), 647--659. https://doi.org/10.2478/s12175-013-0125-2.
[7] J. H. E. Cohn, The Diophantine equation $x^2 + C = y^n$. II, Acta Arith., 109(2) (2003), 205--206.
[8] H. Godinho, M. Diego, A. Togbé, On the Diophantine equation $x^2+ C= y^n$ for $C= 2^a 3^b 17^c$ and $C= 2^a 13^b 17^c$, Math. Slovaca, 66 (2016), 565--574. https://doi.org/10.1515/ms-2015-0159.
[9] S. Gou, T. T. Wang, The Diophantine equation $x^2 + 2^a 17^b = y^n$, Czechoslovak Math. J., 62 (2012), 645--654. https://doi.org/10.1007/s10587-012-0056-z.
[10] X. W. Pan, The exponential Lebesgue–Nagell equation $x^2+p^{2m} = y^n$, Period. Math. Hungar., 67 (2013), 231--242. https://doi.org/10.1007/s10998-013-3044-7.

[11] S. G. Rayaguru, On the Diophantine equation $x^2+C=y^n$, Indian J. Pure Appl. Math., 55 (2024), 69--77. https://doi.org/10.1007/s13226-022-00347-1.
[12] F. S. Abu Muriefah, F. Luca, S. Siksek, Sz. Tengely, On the Diophantine equation $x^2 +C = 2y^n$, Int. J. Number Theory, 5(06) (2009), 1117--1128. https://doi.org/10.1142/S1793042109002572.
[13] P. Baruah, A. Das, A. Hoque, Complete solutions of a Lebesgue–Ramanujan–Nagell type equation, Arch. Math., 60(3) (2024), 135--144. http://dx.doi.org/10.5817/AM2024-3-135.
[14] H. Zhu, M. Le, A. Togbé, On the exponential Diophantine equation $x^2 +p^{2m} = 2y^n$, Bull. Aust. Math. Soc., 86(2) (2012), 303--314. https://doi.org/10.1017/S000497271200010X.
[15] E. K. Mutlu, G. Soydan, On the solutions of some Lebesgue–Ramanujan–Nagell type equations, Int. J. Number Theory, 20(05) (2024), 1195--1218. https://doi.org/10.1142/S1793042124500593.
[16] S. Bhatter, A. Hoque, R. Sharma, On the solutions of a Lebesgue–Nagell type equation, Acta Math. Hungar., 158 (2019), 17--26. https://doi.org/10.1007/s10474-018-00901-6.
[17] K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue–Ramanujan–Nagell equations, Publ. Math. Debrecen, 97 (2020), 339--352.
[18] F. Luca, Sz. Tengely, A. Togbé, On the Diophantine equation $x^2 +C = 4y^n$, Ann. Sci. Math. Québec, 33(2) (2009), 171--184.
[19] S. A. Arif, A. S. Al-Ali, On the Diophantine equation $ax^2+b^m=4y^n$, Acta Arith., 103(4) (2002), 343--346.
[20] Y. Bugeaud, On some exponential Diophantine equations, Monatsh. Math., 132 (2001), 93--97. https://doi.org/10.1007/s006050170046.
[21] K. Chakraborty, A. Hoque, K. Srinivas, On the Diophantine equation $cx^2+p^{2m}=4y^n$, Results Math., 76 (2021), Article ID 57. https://doi.org/10.1007/s00025-021-01366-w.
[22] M. Le, On the Diophantine equations $d_1x^2+2^{2m}d_2=y^n$ and $d_1x^2+d_2=4y^n$, Proc. Amer. Math. Soc., 118 (1993), 67--70. https://doi.org/10.1090/S0002-9939-1993-1152282-5.
[23] F. S. Abu Muriefah, F. Luca, A. Togbé, On the Diophantine equation $x^2 + 5^a 13^b = y^n$, Glasgow Math. J., 50 (2008), 175--181.
[24] K. Chakraborty, A. Hoque, R. Sharma, On the solutions of certain Lebesgue–Ramanujan–Nagell equations, Rocky Mountain J. Math., 51(2) (2021), 459--471. https://doi.org/10.1216/rmj.2021.51.459.
[25] F. Luca, A. Togbé, On the equation $x^2 + 2^\alpha 13^\beta = y^n$, Colloq. Math., 116 (2009), 139--146.
[26] I. Pink, On the Diophantine equation $x^2 + 2^a 3^b 5^c 7^d = y^n$, Publ. Math. Debrecen, 70(1-2) (2007), 149--166. https://doi.org/10.5486/PMD.2007.3477.
[27] H. Zhu, M. Le, G. Soydan, A. Togbé, On the exponential Diophantine equation $x +2^a p^b = y^n$, Period. Math. Hungar., 70 (2015), 233--247. https://doi.org/10.1007/s10998-014-0073-9.
[28] I. Pink, On the Diophantine equation $x^2 + (p_1 \cdots p_k)^2 = 2y^n$, Publ. Math. Debrecen, 65 (2004), 205--213.
[29] Y. Bilu, G. Hanrot, F. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001), 75--122. https://doi.org/10.1515/crll.2001.080.
[30] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(3-4) (1997), 235--265. https://doi.org/10.1006/jsco.1996.0125.
[31] P. Z. Yuan, On the Diophantine equation $ax^2 + by^2 = ck^n$, Indag. Math., 16(2) (2005), 301--320. https://doi.org/10.1016/S0019-3577(05)80030-8.
[32] D. H. Lehmer, An extended theory of Lucas' functions, Ann. Math., 31(3) (1930), 419--448. https://doi.org/10.2307/1968235.
[33] W. Ward, The intrinsic divisors of Lehmer numbers, Ann. Math., 62(2) (1955), 230--236. https://doi.org/10.2307/1969677.
[34] J. H. E. Cohn, Square Fibonacci Numbers, Exploring University Mathematics, London, (1967), 69--82. https://doi.org/10.1016/B978-0-08-011990-8.50009-5.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Murat Alan ^*
0000-0003-2031-2725
Türkiye

Mustafa Aydın
0000-0001-6333-6474
Türkiye

Early Pub Date

September 23, 2025

Publication Date

September 23, 2025

Submission Date

April 13, 2025

Acceptance Date

September 20, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.33434/cams.1675491

IZ

https://izlik.org/JA87KW52JH

APA

Alan, M., & Aydın, M. (2025). Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences, 8(3), 173-182. https://doi.org/10.33434/cams.1675491

AMA

1.Alan M, Aydın M. Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences. 2025;8(3):173-182. doi:10.33434/cams.1675491

Chicago

Alan, Murat, and Mustafa Aydın. 2025. “Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations”. Communications in Advanced Mathematical Sciences 8 (3): 173-82. https://doi.org/10.33434/cams.1675491.

EndNote

Alan M, Aydın M (September 1, 2025) Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences 8 3 173–182.

IEEE

[1]M. Alan and M. Aydın, “Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations”, Communications in Advanced Mathematical Sciences, vol. 8, no. 3, pp. 173–182, Sept. 2025, doi: 10.33434/cams.1675491.

ISNAD

Alan, Murat - Aydın, Mustafa. “Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations”. Communications in Advanced Mathematical Sciences 8/3 (September 1, 2025): 173-182. https://doi.org/10.33434/cams.1675491.

JAMA

1.Alan M, Aydın M. Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences. 2025;8:173–182.

MLA

Alan, Murat, and Mustafa Aydın. “Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations”. Communications in Advanced Mathematical Sciences, vol. 8, no. 3, Sept. 2025, pp. 173-82, doi:10.33434/cams.1675491.

Vancouver

1.Murat Alan, Mustafa Aydın. Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences. 2025 Sep. 1;8(3):173-82. doi:10.33434/cams.1675491

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..

Abstract

Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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