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On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn

  • Amir Ghadermarzi EMAIL logo
Published/Copyright: May 21, 2019
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Abstract

In this note we find all the solutions to the equation x2 + 2α 3β 19γ = yn in nonnegative unknowns with n ≥ 3 and gcd(x, y) = 1, and nonnegative solutions to x2 + 2α 3β 13γ = yn with n ≥ 3, gcd(x, y) = 1, except when α = 0 and x. β. γ is odd.

MSC 2010: Primary 11D41; 11D61
  1. (Communicated by Milan Paštéka )

Acknowledgement

I am grateful to the anonymous referee who brought a number of errors in an earlier version of this paper to my attention.

References

[1] Abu Muriefah, F. S.—Luca, F.—Togb, A.: On the Diophantine equationx2 + 5a13b = yn, Glasg. Math. J. 50 (2008), 175–181.10.1017/S0017089507004028Search in Google Scholar

[2] Apéry, R.: Sur une equation Diophantienne, C. R. Acad. Sci. Paris 251 (1960), 1263–1264.Search in Google Scholar

[3] Arif, S. A.—Abu Muriefah, F. S.: On the Diophantine equationx2 + 2k = yn, Int. J. Math. Math. Sci. 20 (1997), 299–304.10.1155/S0161171297000409Search in Google Scholar

[4] Arif, S. A.—Abu Muriefah, F. S.: The Diophantine equationx2 + 3m = yn, Int. J. Math. Math. Sci. 21 (1998), 619–620.10.1155/S0161171298000866Search in Google Scholar

[5] Arif, S. A.—Abu Muriefah, F. S.: On the Diophantine equationx2 + 2k = yn II, Arab J. Math. Sci. 7 (2001), 67–71.Search in Google Scholar

[6] Bauer, M.—Bennett, M. A.: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation, Ramanujan J. 6(2) (2002), 209–270.10.1023/A:1015779301077Search in Google Scholar

[7] Bennett, M. A.—Skinner C. M.: Ternary diophantine equations via Galois representations and modular forms, Canad. J. Math. 56(1) (2004), 23–54.10.4153/CJM-2004-002-2Search in Google Scholar

[8] Berczes, A.—Brindza, B.—Hajdu, L.: On the power values of polynomials, Publ. Math. Debrecen 53 (1998), 375–381.10.5486/PMD.1998.1993Search in Google Scholar

[9] Beukers, F.: On the generalized Ramanujan-Nagell equation I, Acta Arith. 38 (1980), 389–410.10.4064/aa-38-4-389-410Search in Google Scholar

[10] Beukers, F.: On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 113–123.10.4064/aa-39-2-113-123Search in Google Scholar

[11] Bilu, Yu.—Hanrot, G.—Voutier, P.: Existence of primitive divisors of Lucas and Lehmer numbers. (With an appendix by M. Mignotte), J. Reine Angew. Math. 539 (2001), 75–122.10.1515/crll.2001.080Search in Google Scholar

[12] Bugeaud, Y.—Mignotte, M.—Siksek, S.: Classical and modular approaches to exponantial Diophantine equations II. The Lebesque-Nagell equation, Compositio Math. 142 (2006), 31–62.10.1112/S0010437X05001739Search in Google Scholar

[13] Bugeaud, Y.—Shorey, T. N.: On the number of solutions of the generalized Ramanujan-Nagell equation, J. Reine Angew. Math. 539 (2001), 55–74.10.1515/crll.2001.079Search in Google Scholar

[14] Cangül, I. N.—Demirci, M.—Luca, F.—Pintér, A.—Soydan, G.: On the Diophantine equationx2 + 2a11b = yn, Fibonacci Quart. 48 (2010), 39–46.Search in Google Scholar

[15] Cangül, I. N.—Demirci, M.—Inam, I.—Luca, F.—Soydan, G.: On the Diophantine equationx2 +2a3b11c = yn, Math. Slovaca 63 (2013), 647–659.10.2478/s12175-013-0125-2Search in Google Scholar

[16] Chowla, S.—Lewis, D. J.—Skolem Th.: The Diophantine equation 2n+2 − 7 = x2 and related problems, Proc. Amer. Math. Soc. 10 (1959), 250—257.10.2307/2033452Search in Google Scholar

[17] Cohn, J. H. E.: The Diophantine equationx2 + 2k = yn, Arch. Math. (Basel) 59 (1992), 341–34410.1007/BF01197049Search in Google Scholar

[18] Cohn, J. H. E.: The Diophantine equationx2 + C = yn, Acta Arith. 65 (1993), 367–381.10.4064/aa-65-4-367-381Search in Google Scholar

[19] Goins, E.—Luca, F.—Togbé, A.: On the Diophantine equationx2 + 2α5β13γ = yn. In: ANTS VIII Proceedings, Lecture Notes in Computer. Sci. 5011, Springer, Berlin, 2008, pp. 430–442.10.1007/978-3-540-79456-1_29Search in Google Scholar

[20] Györy, K.—Pink, I.—Pintér, A.: Power values of polynomials and binomial Thue-Mahler equations, Publ. Math. Debrecen 65 (2004), 341–362.10.5486/PMD.2004.3312Search in Google Scholar

[21] Godinho, H.—Marques, D.—Togbé, A.: On the Diophantine equationx2 + C = yn, forC = 2a3b17candC = 2a13b17c Math. Slovaca 66(3) (2016), 565–574.10.1515/ms-2015-0159Search in Google Scholar

[22] Lang, S.: Elliptic Curves: Diophantine Analysis, Springer-Verlag, Berlin-New York, 1978.10.1007/978-3-662-07010-9Search in Google Scholar

[23] Le, M.: Some exponential Diophantine equations I, J. Number Theory 55(2) (1995), 209–221.10.1006/jnth.1995.1138Search in Google Scholar

[24] Lebesque, V. A.: Sur ľimpossibilit en nombres entierde ľequationxm = y2 + 1, Nouv. Ann. Math. 9 (1850), 178–181.Search in Google Scholar

[25] Luca, F.: On the equationx2 + 2a3b = yn, Int. J. Math. Math. Sci. 29 (2002), 239–244.10.1155/S0161171202004696Search in Google Scholar

[26] Luca, F.—Togbé, A.: On the Diophantine equationx2 + 2a5b = yn, Int. J. Number Theory 4(6) (2008), 973–979.10.1142/S1793042108001791Search in Google Scholar

[27] Luca, F.—Togbé, A.: On the Diophantine equationx2 + 72k = yn, Fibonacci Quart. 54 (2007), 322–326.Search in Google Scholar

[28] Mahler, K.: Lectures on Diophantine Approximations, Univ. of Notre Dame, Notre Dame, IN, 1961.Search in Google Scholar

[29] Mignotte, M.—De Weger, B. M. M.: On the equationsx2 +74 = y5 and x2 + 86 = y5, Glasg. Math. J. 38 (1996), 77–85.10.1017/S0017089500031293Search in Google Scholar

[30] Nagell, T.: Sur ľimpossibilité de quelques équations a deux indeterminées, Norsk. Mat. Forensings Skifter 13 (1923), 65–82.Search in Google Scholar

[31] Nagell, T.: L⊘sning till oppgave nr 2, Nordisk Mat. Tidskr 30 (1948), 62–64.Search in Google Scholar

[32] Pethö, A.—Zimmer, H. G.—Gebel, J.—Hermann, E.: Computing allS-integral points on elliptic curves, Math. Proc. Cambridge Phil. Soc. 27 (1999), 383–402.10.1017/S0305004199003916Search in Google Scholar

[33] Pink, I.: On the Diophantine equationx2 + 2a3b5c7d = yn, Publ. Math. Debrecen 70 (2007), 149–166.10.5486/PMD.2007.3477Search in Google Scholar

[34] Ramanujan, S.: Question 446, J. Indian Math. Soc. 5 (1913), 120, Collected papers, Cambridge University Press (1927), 327.Search in Google Scholar

[35] Shorey, T. N.—Van Der Poorten, A. J.—Tijdeman, R.—Schinzel, A.: Applications of the Geľfond-Baker method to Diophantine quations. In: Transcendence Theory: Advances and Applications, Academic Press, London-New York, San Francisco, 1977, pp. 59–77.Search in Google Scholar

[36] Smart, N.P. The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, 1998.10.1017/CBO9781107359994Search in Google Scholar

[37] Soydan, G.—Ulas, M.—Zhu, H.: On the Diophantine equationx2 + 2a19b = yn, Indian J. Pure Appl. Math. 43 (2012), 251–261.10.1007/s13226-012-0013-4Search in Google Scholar

[38] Tao, L.: On the Diophantine equationx2 + 5m = yn, Ramanujan J. 19 (2009), 325–338.10.1007/s11139-008-9152-ySearch in Google Scholar

[39] Zagier, D.: Large integral points on elliptic curves, Math. Comp. 48 (1987), 425–436.10.1090/S0025-5718-1987-0866125-3Search in Google Scholar

Received: 2017-08-25
Accepted: 2018-10-23
Published Online: 2019-05-21
Published in Print: 2019-06-26

© 2019 Mathematical Institute Slovak Academy of Sciences

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