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Year 2025, Volume: 8 Issue: 3, 173 - 182, 23.09.2025

Murat Alan , Mustafa Aydın

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

Abstract

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.

Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations

Year 2025, Volume: 8 Issue: 3, 173 - 182, 23.09.2025

Murat Alan , Mustafa Aydın

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

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.
There are 34 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Research Article
Authors

Murat Alan 0000-0003-2031-2725

Mustafa Aydın 0000-0001-6333-6474

Submission Date April 13, 2025
Acceptance Date September 20, 2025
Early Pub Date September 23, 2025
Publication Date September 23, 2025
Published in Issue Year 2025 Volume: 8 Issue: 3

Cite

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 Alan M, Aydın M. Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences. September 2025;8(3):173-182. doi:10.33434/cams.1675491
Chicago Alan, Murat, and Mustafa Aydın. “Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations”. Communications in Advanced Mathematical Sciences 8, no. 3 (September 2025): 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 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, 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 (September2025), 173-182. https://doi.org/10.33434/cams.1675491.
JAMA 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, 2025, pp. 173-82, doi:10.33434/cams.1675491.
Vancouver Alan M, Aydın M. Concerning the Solutions of Some Lebesgue-Ramanujan-Nagell Equations. Communications in Advanced Mathematical Sciences. 2025;8(3):173-82.

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