TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

An exponential Diophantine equation x2 + 3a 73b = yn

, *

* Corresponding author · click or hover a name for details

pp. 2289–2296Vol. 28Issue 6September 2025DOI: 10.47974/JDMSC-2225 Crossmark XML
Received:
03 Dec 2024
Published Online:
16 Sep 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2225
Pages:
2289–2296

Abstract

This paper aims to identify all solutions in positive integers x and y (where x, y ≥ 1), with nonnegative exponents a and b, and an integer n ≥ 3, that satisfy the Diophantine equation x2 + 3a 73b = yn under the condition x and y are coprime.

Keywords

Subject Classifications

11D6111D4111Y50

References

[1] F. S. Abu Muriefah and S. A. Arif, “The Diophantine equation x2 + 3m = yn,” Int. J. Math. Math. Sci., vol. 21, no. 3, pp. 619-620 (1998).
[2] F.S. Abu Muriefah and S.A. Arif, “The Diophantine equation x2 + 52k+1 = I,” Indian J. Pure Appl. Math., vol. 30, no. 3, pp. 229–231 (1999).
[3] F.S. Abu Muriefah and S.A. Arif, “On the Diophantine equation x2 + 2k = yn,” Int. J. Math. Math. Sci., vol. 20, no. 2, pp. 299-304 (1997).
[4] F.S. Abu Muriefah, “On the Diophantine equation x2 + 52k = yn,” Demonstratio Math., vol. 39, no. 2, pp. 285-289 (2006).
[5] M. Alan and U. Zengin, “On the Diophantine equation x2 + 3a 41b = yn,” Period. Math. Hung., vol. 81, pp. 284-291 (2020).
[6] S.A. Arif and F.S. Abu Muriefah, “On the Diophantine equation x2 + 2k = yn II,” Arab J. Math. Sci., vol. 7, no. 1, pp. 67–71 (2001).
[7] S.A. Arif  and F.S. Abu Muriefah, “On the Diophantine equation x2 + q2k+1 = yn,” J. Number Theory, vol. 95, pp. 95–100 (2002).
[8] Y. Bilu, G. Hanrot and P.M. Voutier, “Existence of primitive divisors of Lucas and Lehmer numbers (with Appendix by Mignotte),” J. Reine Angew. Math., vol. 539, pp. 75-122 (2001).
[9] A. Bérczes and I. Pink, “On the Diophantine equation x2 + p2k = yn,” Arch. Math., vol. 91, pp. 505–517 (2008).
[10] A. Bérczes and I. Pink, “On generalized Lebesgue–Ramanujan–Nagell equation,” An. St. Univ. Ovid. Cons., vol. 22, no. 1, pp. 51-57 (2014).
[11] Y. Bugeaud, M. Mignotte and S. Siksek, “Classical and modular approaches to exponential Diophantine equations II, The Lebesgue–Nagell equation,” Compos. Math., vol. 142, pp. 31–62 (2006).
[12] W. Bosma, J. Cannon and C. Playoust, “The Magma algebra system. I. The user language,” J. Symbolic Comput., vol. 24, no. 3, pp. 235–265 (1997).
[13] I.N. Cangúl, M. Demirci, F. Luca, Á. Pintér and G. Soydan, “On the Diophantine equation x2 + 2a 11b =  yn,” Fibonacci Q., vol. 48, no. 1, pp. 39–46 (2010).
[14]  I.N. Cangúl, M. Demirci, M. Inam, F. Luca and G. Soydan, “On the Diophantine equation x2 + 2a 3b 11c = yn, Math. Slovaca, vol. 63, no. 3, pp. 647–659 (2013).
[15] R.D. Carmichael, “On the numerical factors of the arithmetic forms αn-βn,” Ann. Math., vol. 2, no. 15, pp. 30–70 (1913).
[16] J.H.E. Cohn, “The Diophantine equation x2 + 2k = yn,” Arch. Math. Basel, vol. 59, no. 4, pp. 341–344 (1992).
[17] J.H.E. Cohn, “The Diophantine equation x2 + C = yn,” Acta Arith., vol. 65, no. 4, pp. 367–381 (1993).
[18] M. Demirci, “On the Diophantine equation x2 + 5a  pb = yn,” Filomat, vol. 31, no. 16, pp. 5263–5269 (2017).
[19] H. Godinho, D. Marques and A. Togbé, “On the Diophantine equation x2 + 2α  5β 17γ = yn,” Commun.Math., vol. 20, no. 2, pp. 81–88 (2012).
[20] H. Godinho, D. Marques and A. Togbé, “On the Diophantine equation x2 + C = yn for C = 2a 3b 17c and C = 2a 13b 17c,” Math. Slovaca, vol. 66, no. 4, pp. 565–574 (2016).
[21] C. Ko, “On the Diophantine equation x2 = yn+1, xy ≠ 0,” Sci.Sin., vol. 14, pp. 457–460 (1965).
[22] F. Landau and A. Ostrowski, “On the Diophantine equation ay2 + by + c = dxn,” Proc. Lond. Math. Soc., vol. 19, no. 2, pp. 276–280 (1920).
[23] M. Le, “An exponential Diophantine equation,” Bull. Austral. Math. Soc., vol. 64, pp. 99–105 (2001).
[24] L.A. Lebesgue, “Sur l’impossibilité, en nombres entiers, de l’équation xm = y2  + 1,” Nouv. Ann. Math., vol. 9, no. 1, pp. 178–181 (1850).
[25] F. Luca, “On the equation x2 + 2a 3b = yn,” Int. J. Math. Math. Sci., vol. 29, no. 3, pp. 239–244 (2002).
[26] F. Luca and A. Togbé, “On the equation x2 + 2a 5b = yn,” Int.J.Number Theory, vol. 4, no. 6, pp. 973–979 (2008).
[27] F. Luca and A. Togbé, “On the equation x2 + 2α 13β = yn,” Colloq. Math., vol. 116, no. 1, pp. 139–146 (2009).
[28] F. Luca and A. Togbé, “On the Diophantine equation x2 + 72k = yn,” Fibonacci Quart., vol. 45, no. 4, 322–326 (2007).
[29] M. Mignotte and B.M.M. DeWeger, “On the Diophantine equations x2 + 74 = y5 and x2 + 86 = y5,” Glasgow Math. J., vol. 38, no. 1, pp. 77–85 (1996).
[30] S. Muthuvel and R. Venkatraman, “An Exponential Diophantine Equation x2 + 3a 97b = yn,” Int. J. Math. Comput. Sci., vol. 19, no. 4, pp. 1125-1128 (2024).
[31] S. Muthuvel and R. Venkatraman, “An Exponential Diophantine Equation x2 + 3a 113b = yn,” arXiv:2405.09252v1.
[32] S. Muthuvel and R. Venkatraman, “An Exponential Diophantine Equation x2 + 3a 89b = yn,” Palestine J. Math., vol. 14, special issue II, pp. 62-67 (2025).
[33] T. Nagell, “Sur l’impossibilité en nombres entiers de quelques équations a deux indéterminés,” Norsk. Mat. Forensings Skifter Ser. I, vol. 13, no. 1, pp. 65–82 (1923).
[34] T. Nagell, “Contributions to the theory of a category of Diophantine equations of the second degree with two unknowns,” Nova Acta Reg. Soc. Upsal. IV Ser., vol. 2, no. 15, pp. 1–38 (1955).
[35] X. Pan, “The exponential Lebesgue-Nagell equation x2 + p2m = yn,” Period. Math. Hung., vol. 67, no. 2, pp. 231–242 (2013).
[36] I. Pink, “On the Diophantine equation x2 + 2a 3b 5c 7d = yn,” Publ. Math. Debrecen, vol. 70, no. (1–2), pp. 149–166 (2007).
[37] I. Pink, Z. Rabai, “On the Diophantine equation x2 + 5k 17l = yn,” Commun. Math., vol. 19, pp. 1–9 (2011).
[38] S. G. Rayaguru, “On the Diophantine equation x2 + C = yn,” Indian J. Pure Appl. Math., pp. 1-9, 2022.
[39] G. Soydan, M. Ulas and H. Zhu, “On the Diophantine equation x2 + 2a 19b  = yn,” Indian J. Pure Appl. Math., vol. 43, no. 3, pp. 251–261 (2012).
[40] G. Soydan and N. Tzanakis, “Complete solution of the Diophantine equation x2 + 5a 11b = yn,” Bull. Hellenic Math Soc., vol. 60, pp. 125–152 (2016).
[41] H. Zhu, “A note on the Diophantine equation x2 + qm = y3,” Acta Arith., vol. 146, no. 2, pp. 195–202 (2011).
[42] H. Zhu, M. Le, G. Soydan and A. Togbé, “On the exponential Diophantine equation x2 + 2apb = yn,” Period. Math. Hung., vol. 70, pp. 233–247 (2015).

Views: 97Downloads: 5Citations: 0