On Generalized Lebesgue-Ramanujan-Nagell Equations

Attila Bérczes; István Pink

doi:10.2478/auom-2014-0006

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On Generalized Lebesgue-Ramanujan-Nagell Equations

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 22 (2014): Issue 1 (March 2014)

By: Attila Bérczes and István Pink

Open Access

|Dec 2014

Abstract

We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x² + 11^a 17^b = yⁿ in nonnegative integer unknowns with n ≧ 3 and gcd(x, y) = 1.

References

[1] F.S. Abu Muriefah, On the Diophantine equation x² + 5^2k = yⁿ, Demonstratio Mathematica 319/2 (2006), 285–289.10.1515/dema-2006-0206
Search in Google Scholar
[2] F.S. Abu Muriefah, F. Luca, S. Siksek, and Sz. Tengely, On the Diophantine equation x² + C = 2yⁿ, Int. J. Number Theory 5 (2009), 1117–1128.10.1142/S1793042109002572
Search in Google Scholar
[3] R. Apéry, Sur une equation diophantienne (French), C. R. Acad. Sci. Paris 251 1960, 1263-1264.
Search in Google Scholar
[4] R. Apéry, Sur une equation diophantienne (French), C. R. Acad. Sci. Paris 251 1960, 1451-1452.
Search in Google Scholar
[5] S.A. Arif and F.S.A. Muriefah, On the Diophantine equation x² + 2^k = yⁿ, Internat. J. Math. Math. Sci. 20 (1997), 299–304.10.1155/S0161171297000409
Search in Google Scholar
[6] S.A. Arif and F.S.A. Muriefah, The Diophantine equation x² + 3^m = yⁿ, Internat. J. Math. Math. Sci. 21 (1998), 619–620.10.1155/S0161171298000866
Search in Google Scholar
[7] S.A. Arif and F.S.A. Muriefah, The Diophantine equation x² + q^2k = yⁿ, Arab. J. Sci. Sect. A Sci. 26 (2001), 53–62.
Search in Google Scholar
[8] S.A. Arif and F.S.A. Muriefah, On the Diophantine equation x² + 2^k = yⁿ II, Arab J. Math. Sci. 7 (2001), 67–71.
Search in Google Scholar
[9] S.A. Arif and F.S.A. Muriefah, On the Diophantine equation x² + q^2k+1 = yⁿ, J. Number Theory 95 (2002), 95–100.10.1006/jnth.2001.2750
Search in Google Scholar
[10] M. Bauer and M. A. Bennett, Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation, Ramanujan J. 6 (2002), 209-270.10.1023/A:1015779301077
Search in Google Scholar
[11] E. A. Bender, N. P. Herzberg, Some Diophantine equations related to the quadratic form ax²+by², in: Studies in Algebra and Number Theory, Academic Press, New York 1979, 219-272.
Search in Google Scholar
[12] M.A. Bennett, J.S. Ellenberg and N. Ng, The Diophantine equation A⁴ + 2^dB² = Cⁿ Inter. J. Number Theory 6 (2010), 1–27.
Search in Google Scholar
[13] M.A. Bennett and C.M. Skinner, Ternary diophantine equations via Galois representations and modular forms, Canad. J. Math. 56/1, (2004), 23–54.10.4153/CJM-2004-002-2
Search in Google Scholar
[14] A. Bérczes, B. Brindza and L. Hajdu, On power values of polynomials, Publ. Math. Debrecen, 53, (1998), 375–381.10.5486/PMD.1998.1993
Search in Google Scholar
[15] A. Bérczes and I. Pink, On the diophantine equation x²+p^2k = yⁿ, Archiv der Mathematik, 91 (2008), 505-517.10.1007/s00013-008-2847-x
Search in Google Scholar
[16] A. Bérczes and I. Pink, On the Diophantine equation x² +d^2l+1 = yⁿ, Glasg. Math. Journal, 54 (2012), 415-428.10.1017/S0017089512000067
Search in Google Scholar
[17] F. Beukers, On the generalized Ramanujan-Nagell equation I, Ac-tha Arith. 38 (1980/1981), 389-410.10.4064/aa-38-4-389-410
Search in Google Scholar
[18] F. Beukers, On the generalized Ramanujan-Nagell equation II, Ac-tha Arith. 39 (1981), 113-123.10.4064/aa-39-2-113-123
Search in Google Scholar
[19] Y. Bilu, G. Hanrot and P.M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers. With an appendix by M. Mignotte., J. Reine Angew. Math. 539 (2001), 75–122.
Search in Google Scholar
[20] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235– 265.10.1006/jsco.1996.0125
Search in Google Scholar
[21] J. Browkin and A. Schinzel, On the equation 2ⁿ - D = y², Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 8 (1960), 311-318.
Search in Google Scholar
[22] Y. Bugeaud On the diophantine equation x² - p^m = ±yⁿ, Acta Arith. 80, (1997), 213–223.10.4064/aa-80-3-213-223
Search in Google Scholar
[23] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential and diophantine equations II. The Lebesque-Nagell equation, Compos. Math. 142/1 (2006), 31–62.10.1112/S0010437X05001739
Search in Google Scholar
[24] Y. Bugeaud and F.S. Abu Muriefah, The Diophantine equation x²+c = yⁿ: a brief overview, Revista Colombiana de Matematicas, 40, (2006), 31–37.
Search in Google Scholar
[25] Y. Bugeaud and T.N. Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation, J. reine angew. Math. 539 (2001), 55–74.
Search in Google Scholar
[26] I.N. Cangül, M. Demirci, G. Soydan, N. Tzanakis, On the Dio-phantine equation x² + 5^a11^b = yⁿ, Funct. Approx. Comment. Math. 43 (2010), 209–225.
Search in Google Scholar
[27] I. N. Cangül, M. Demirci, F. Luca, A. Pintér and G. Soydan, On the Diophantine equation x² + 2^a11^b = yⁿ, Fibonacci Quart. 48 (2010), 39-46.
Search in Google Scholar
[28] I. N. Cangül, M. Demirci, I. Inam, F. Luca and G. Soydan, ON THE DIOPHANTINE EQUATION x² +2^a3^b11^c = yⁿ, Math. Slo-vaca 63 (2013), 647–659.10.2478/s12175-013-0125-2
Search in Google Scholar
[29] I. N. Cangül, G. Soydan and Y. Simsek, A p-adic Look at the Dio-phantine Equation x² + 11^2k = yⁿ, AIP Conf. Proc. 1168 (2011), 275-277.
Search in Google Scholar
[30] S. Cerberci and H. Senay, The Diophantine Equation x² +q^m = pⁿ, Int. J. Contemp. Math. Sciences 24 (2009), 1181-1191.
Search in Google Scholar
[31] S. Cenberci and B. Peker, On the solutions of the equation x² + 19^m = yⁿ, Notes on Number Theory and Discrete Mathematics 18 (2012), 34-41.
Search in Google Scholar
[32] S. Chowla, D. J. Lewis and Th. Skolem, The Diophantine equation 2n+2 _ 7 = x2 and related problems, Proc. Amer. Math. Soc. 10 (1959), 250-257.
Search in Google Scholar
[33] J.H.E. Cohn, The diophantine equation x² + 2^k = yⁿ.II., Int. J. Math. Math. Sci., 22, (1999), 459-462.10.1155/S0161171299224593
Search in Google Scholar
[34] J.H.E. Cohn, The diophantine equation x² + 2^k = yⁿ, Arch. Math (Basel) 59 (1992), 341-344.10.1007/BF01197049
Search in Google Scholar
[35] J.H.E. Cohn, The diophantine equation x² + C = yⁿ, Acta Arith. 65 (1993), 367-381.10.4064/aa-65-4-367-381
Search in Google Scholar
[36] J.H.E. Cohn, The diophantine equation x² + C = yⁿ II, Acta Arith. 109 (2003), 205-206.10.4064/aa109-2-8
Search in Google Scholar
[37] E. Demirpolat, S. Cerberci and H. Senay, The Diophantine Equation x² + 11^2k+1 = yⁿ, International Mathematical Forum, 4, no. 6 (2009), 277-280.
Search in Google Scholar
[38] J. S. Ellenberg, Galois representations to Q-curves and the generalized Fermat Equation A⁴ + B² = C^p, Amer. J. Math. 126, 763-787 (2004).10.1353/ajm.2004.0027
Search in Google Scholar
[39] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463-471.
Search in Google Scholar
[40] H. Godinho, D. Marques and A. Togbe, On the Diophantine equation x² + 2^α5^β17^γ = yⁿ, Communications in Mathematics 20 (2012) 81-88.
Search in Google Scholar
[41] E. Goins, F. Luca and A. Togbe, On the Diophantine Equation x² + 2^α5^β13^γ = yⁿ, ANTS VIII Proceedings: A.J. van der Poorten and A. Stein (eds.), ANTS VIII, Lecture Notes in Computer Science 5011 (2008), 430-442.10.1007/978-3-540-79456-1_29
Search in Google Scholar
[42] K. Györy, I. Pink and Á. Pintér, Power values of polynomials and binomial Thue-Mahler equations, Publ. Math. Debrecen 65 (2004), 341-362.10.5486/PMD.2004.3312
Search in Google Scholar
[43] H. Hasse, Über eine diophantische Gleichung von Ramanujan-Nagell und ihre Verallgemeinerung, Nagoya Math. J. 27 (1966), 77-102.10.1017/S0027763000011892
Search in Google Scholar
[44] Y. Hu and M. Le, New Advances on the Generalized Lebesgue-Ramanujan-Nagell Equation, Advances in Mathematics(China) 41 (2012), 385-396.
Search in Google Scholar
[45] M. Le, On the number of solutions of the generalized Ramanujan-Nagell equation x² – D = 2ⁿ+², Acta Arith. 60 (1991), 149-167.10.4064/aa-60-2-149-167
Search in Google Scholar
[46] M. Le, A note on the Diophantine Equation x² + 4D = y^p, Monatsh. Math. 116 (1993), 283-285.10.1007/BF01301534
Search in Google Scholar
[47] M. Le, On the number of solutions of the diophantine equation x² + D = pⁿ, C. R. Acad. Sci. Paris Ser. I. Math. 317 (1993), 135-138.
Search in Google Scholar
[48] M. Le, On the Diophantine equation D1x² + D2 = 2ⁿ⁺², Acta Arith. 64 (1993), 29-41.10.4064/aa-64-1-29-41
Search in Google Scholar
[49] M. Le, A note on the Generalized Ramanujan-Nagell Equation, J. of Number Theory 50 (1995), 193-201.10.1006/jnth.1995.1013
Search in Google Scholar
[50] M. Le, Some Exponential Diophantine Equations I. The Equation D₁x² – D_2y² = λk^z, J. Number Theory 55 (1995), 209-221.10.1006/jnth.1995.1138
Search in Google Scholar
[51] M. Le, A Note on the Number of Solutions of the Generalized Ramanujan-Nagell Equation D_1x² + D₂ = 4pⁿ, J. of Number Theory 62 (1997), 100-106.10.1006/jnth.1997.2019
Search in Google Scholar
[52] M. Le, On the Diophantine equation (x³ – 1)/(x – 1) = (yⁿ – 1)/(y – 1), Trans. Amer. Math. Soc. 351 (1999), 1063-1074.10.1090/S0002-9947-99-02013-9
Search in Google Scholar
[53] M. Le, On Cohn's conjecture concerning the diophantine equation x² + 2^m = yⁿ, Arch. Math. Basel 78 (2002), 26-35.10.1007/s00013-002-8213-5
Search in Google Scholar
[54] M. Le, On the diophantine equation x² + p² = yⁿ, Publ. Math. Debrecen 63 (2003), 27-78.10.5486/PMD.2003.2636
Search in Google Scholar
[55] M. Le and H. Zhu, On some generalized Lebesque-Nagell equations, Journal of Number Theory 131 (2011), 458-469.10.1016/j.jnt.2010.09.009
Search in Google Scholar
[56] V. A. Lebesque, Sur l'impossibilité en nombres entierde l'equation x^m=y²+1I Nouvelle Annales des Mathématiques (1) 9 (1850), 178–181.
Search in Google Scholar
[57] A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.10.1007/BF01457454
Search in Google Scholar
[58] M.G. Leu and G.W. Li The Diophantine equation 2x² + 1 = 3ⁿ, Proc. Amer. Math. Soc. 131 (2003) 3643-3645.10.1090/S0002-9939-03-07212-5
Search in Google Scholar
[59] W. Ljunggren, Über einige Arcustangensgleichungen die auf in-teressante unbestimmte Gleichungen führen, Ark. Mat. Astr. Fys. 29A (1943), No. 13.
Search in Google Scholar
[60] W. Ljunggren, on the diophantine equationCx²+D =yⁿ, Pacific J. Math. 14 (1964), 585–596.10.2140/pjm.1964.14.585
Search in Google Scholar
[61] F. Luca, On aIdiophantine equation, Bull. Austral. Math. Soc. 61 (2000), 241–246.10.1017/S0004972700022231
Search in Google Scholar
[62] F. Luca, On the equation x² + 2^a3^b = yⁿ, Int. J. Math. Sci. 29 (2002), 239–244.10.1155/S0161171202004696
Search in Google Scholar
[63] F. Luca, Sz. Tengely, and A. Togbe, On the Diophantine equation x² + C = 4yⁿ, Ann. Sci. Math. Quebec 33 (2009), 171–184.
Search in Google Scholar
[64] F. Luca and A. Togbe, On the Diophantine equation x²+7^2k = yⁿ, Fibonacci Quart. 54/4 (2007), 322-326.
Search in Google Scholar
[65] F. Luca and A. Togbe, On the Diophantine equation x² + 2^a5^b = yⁿ, Int. J. Number Th. 4/6 (2008), 973-979.10.1142/S1793042108001791
Search in Google Scholar
[66] M. Mignotte and B.M.M de Weger, On the equations x² +74 = y⁵ and x² + 86 = y⁵, Glasgow Math. J. 38/1 (1996), 77–85.10.1017/S0017089500031293
Search in Google Scholar
[67] R. A. Mollin, A note on the Diophantine Equation D₁x² + D₂ = akⁿ, Acta Math. Acad. Paedagog. Nyhzi (N.S.) 21 (2005), 21-24.
Search in Google Scholar
[68] F.S. Abu Muriefah, F. Luca and A. Togbe, On the diophantine equation x² + 5^a13^b = yⁿ, Glasgow Math. J 50 (2008), 175–181.10.1017/S0017089507004028
Search in Google Scholar
[69] T. Nagell, Sur l'impossibilité de quelques équations a deux indeterminées, Norsk. Mat. Forensings Skifter 13 (1923), 65–82.
Search in Google Scholar
[70] T. Nagell, L⊘sning till oppgave nr 2, Norsk. Mat. Tidsskrift 30 (1948), 62–64.
Search in Google Scholar
[71] 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. 16, Uppsala 1955, pp. 1-38.
Search in Google Scholar
[72] T. Nagell The Diophantine equation x² + 7 = 2ⁿ, Ark. Math. 4 (1960), 185-187.10.1007/BF02592006
Search in Google Scholar
[73] A. Pethö, H.G. Zimmer, J. Gebel and E. Hermann Computing all S-integral points on elliptic curves, Math. Proc. Cambridge. Phil. Soc 127 (1999) 383-402.10.1017/S0305004199003916
Search in Google Scholar
[74] I. Pink On the diophantine equation x² + 2^α3^β5^γ7^δ = yⁿ, Publ. Math. Debrecen 70/1-2 (2007),149-166.10.5486/PMD.2007.3477
Search in Google Scholar
[75] I. Pink and Zs. Rábai On the diophantine equation x² +5^k17^l = yⁿ, Communications in Mathematics, 19, (2011), 1-9.
Search in Google Scholar
[76] S. Ramanujan, Question 446, J. Indian Math. Soc. 5 (1913), 120, Collected papers, Cambridge University Press (1927), 327.
Search in Google Scholar
[77] N. Saradha and A. Srinivasan Solutions of some generalized Ramanujan-Nagell equations, Indag. Math. (N.S.) 17/1 (2006), 103-114.10.1016/S0019-3577(06)80009-1
Search in Google Scholar
[78] N. Saradha and A. Srinivasan Solutions of some generalized Ramanujan-Nagell equations via binary quadratic forms, Publ. Math. Debrecen 71/3-4 (2007), 349-374.10.5486/PMD.2007.3735
Search in Google Scholar
[79] N. Saradha and A. Srinivasan, Generalized Lebesgue-Ramanujan-Nagell Equations, (2008), 207-223, Diophantine Equations, Editor: N. Saradha, Narosa Publishing House, New Delhi, India.
Search in Google Scholar
[80] A. Schinzel and R. Tijdeman, On the equation y^m = P(x), Acta Arith. (1976), 31, 199-204.10.4064/aa-31-2-199-204
Search in Google Scholar
[81] T.N. Shorey, A.J. van der Poorten, R. Tijdeman and A. Schinzel, Applications of the Gel'fond-Baker method to Diophantine equations, in: Transcendence Theory: Advances and Applications, Academic Press, London-New York, San Francisco, (1977), 59-77.
Search in Google Scholar
[82] T.N. Shorey, R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, 87. Cambridge University Press, Cambridge, 1986, X+240 pp.10.1017/CBO9780511566042
Search in Google Scholar
[83] N. P. Smart, Determining the small solutions to S-unit equations, Math. Comp. 68 (1999), 1687-1699.10.1090/S0025-5718-99-01140-0
Search in Google Scholar
[84] G. Soydan, On the Diophantine equation x² +7^α 11^β = yⁿ, Miskolc Mathematical Notes 13 (2012), 515-527.10.18514/MMN.2012.424
Search in Google Scholar
[85] G. Soydan, M. Ulas and H. Zhu, ON THE DIOPHANTINE EQUATION x² + 2^a19^b = yⁿ, Indian J. Pure Appl. Math. 43 (2012), 251-261.10.1007/s13226-012-0013-4
Search in Google Scholar
[86] R. J. Stroeker and N. Tzanakis Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith-metica 67 (1994), 177-196.10.4064/aa-67-2-177-196
Search in Google Scholar
[87] L. Tao, On the Diophantine equation X² + 3^m = Yⁿ, Integers: Electronic J. Combinatorial Number Theory 8 (2008), 1-7.
Search in Google Scholar
[88] L. Tao, On the Diophantine equation x² + 5^m = yⁿ, Ramanujan J. 19 (2009), 325-338.10.1007/s11139-008-9152-y
Search in Google Scholar
[89] Sz. Tengely, On the Diophantine equation x² + a² = 2y^p, Indag. Math. (N.S.) 15 (2004), 291-304.10.1016/S0019-3577(04)90021-3
Search in Google Scholar
[90] Sz. Tengely, On the Diophantine equation x² + q^2m = 2y^p, Acta Arith. 127 (2007), 71-86.10.4064/aa127-1-6
Search in Google Scholar
[91] H. Virgolici, On the Exponential Diophantine Equation x² + D = yⁿ: a brief survey, An. Univ. Spiru Haret. Ser. Mat.-Inform. 9 (2013), 45-54.
Search in Google Scholar
[92] B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract 65, Stichting Mathematisch Centrum, Amsterdam 1989.
Search in Google Scholar
[93] K. Wildanger, Über das Lösen von Einheiten- und Indexformgle-ichungen in algebraischen Zahlkrrpern, J. of Number Theory 82 (2000), 188-224.10.1006/jnth.1999.2414
Search in Google Scholar
[94] P. Xiaowei, The Exponential Lebesgue-Nagell Equation x² +p^2m = yⁿ, Period. Math. Hungar. 67 (2013), 231-242.10.1007/s10998-013-3044-7
Search in Google Scholar
[95] P. Yuan, On the number of the solutions of x² – D = pⁿ, Sichuan Daxue Xuebao 35 (1998), 311-316.
Search in Google Scholar
[96] H. Zhu, A note on the Diophantine equation x² + q^m = y³, Acta Arith. 146 (2011), 195-202.10.4064/aa146-2-6
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/auom-2014-0006 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 51 - 71

Submitted on: Jan 1, 2014

Accepted on: Feb 1, 2014

Published on: Dec 10, 2014

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Generalized Lebesgue-Ramanujan-Nagell equations,

Primitive divisor theorem

Related subjects:

Mathematics,

General mathematics

© 2014 Attila Bérczes, István Pink, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 22 (2014): Issue 1 (March 2014)