Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
-
Yasutsugu Fujita
and
Maohua Le
Abstract
Let D be a nonsquare integer, and let k be an integer with |k| ≥ 1 and gcd(D, k) = 1. In the part I of this paper, using some properties on the representation of integers by binary quadratic primitive forms with discriminant 4D, M.-H. Le gave a series of explicit formulas for the integer solutions (x, y, z) of the equation x2 – Dy2 = kz, gcd(x, y) = 1, z > 0 for 2 ∤ k or |k| is a power of 2. In this part, we give similar results for the other cases of k.
Acknowledgement
The authors thank the referees for their careful reading and pertinent suggestions.
-
Communicated by István Gaál
References
[1] Abu Muriefah, F. S.—Bugeaud, Y.: The Diophantine equationx2 + C = yn: a brief overview, Rev. Colomb. Math. 40 (2006), 31–37.10.1515/dema-2006-0206Search in Google Scholar
[2] Bérczes, A.—Pink, I.: On generalized Lebesgue-Ramanujan-Nagell equations, An. St. Univ. Ovidius Constanta 22 (2014), 51–71.10.2478/auom-2014-0006Search in Google Scholar
[3] Bugeaud, Y.: On the Diophantine equationx2 – pm = ±yn, Acta Arith. 80 (1997), 213–223.10.4064/aa-80-3-213-223Search in Google Scholar
[4] 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
[5] Gou, S.—Wang, T.: The Diophantine equationx2 + 2a17b = yn, Czechoslovak Math. J. 62 (2012), 645–654.10.1007/s10587-012-0056-zSearch in Google Scholar
[6] Hua, L.-K.: Introduction to Number Theory, Berlin, Springer-Verlag, 1982.Search in Google Scholar
[7] Landau, E.: Vorlesungen uber Zahlentheorie, Leipzig, Hirzel, 1927.Search in Google Scholar
[8] Le, M.-H.: On the representation of integers binary quadratic primitive forms I, J. Changchun Teachers College, Nat. Sci. 3 (1986), 3–12 (in Chinese).Search in Google Scholar
[9] Le, M.-H.: On the representation of integers binary quadratic primitive forms II, J. Changsha Railway Institute 7 (1989), 6–18 (in Chinese).Search in Google Scholar
[10] Le, M.-H.: Some exponential Diophantine equations I: The equationD1x2 – D2y2 = λkz, J. Number Theory 55 (1995), 209–221.10.1006/jnth.1995.1138Search in Google Scholar
[11] Le, M.-H.—Hu, Y.-Z.: New advances on generalized Lebesgue-Ramanujan-Nagell equation, Adv. Math. Beijing 41 (2012), 385–397 (in Chinese).Search in Google Scholar
[12] Le, M.-H.—Soydan, G.: A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation, Surv. Math. Appl. 15 (2020), 473–523.Search in Google Scholar
[13] Mordell, L. J.: Diophantine Equations, London, Academic Press, 1969.Search in Google Scholar
[14] Pink, I.: On the Diophantine equationx2+2α3β5γ7δ = yn, Publ. Math. Debrecen 70 (2007), 149–166.10.5486/PMD.2007.3477Search in Google Scholar
[15] Saradha, N.—Srinivasan, A.: Solutions of some generalized Ramanujan-Nagell equations via binary quadratic forms, Publ. Math. Debrecen 71 (2007), 349–374.10.5486/PMD.2007.3735Search in Google Scholar
[16] WANG, J.-P.—WANG, T.-T.—ZHANG, W.-P.: The exponential Diophantine equationx2+(3n2 + 1)y = (4n2 + 1)z, Math. Slovaca 64 (2014), 1145–1152.10.2478/s12175-014-0265-zSearch in Google Scholar
[17] Yang, H.—Fu, R.: An upper bound for least solutions of the exponential Diophantine equationD1x2 – D2y2 = λkz, Int. J. Number Theory 11 (2015), 1107–1114.10.1142/S1793042115500608Search in Google Scholar
[18] Yang, H.—Fu, R.: The exponential diophantine equation xy + yx = z2 via a generalization of the Ankeny-Artin-Chowla conjecture, Int. J. Number Theory 14 (2018), 1223–1228.10.1142/S1793042118500756Search in Google Scholar
[19] Yang, H.—Fu, R.: A note on the Goormaghtigh equation, Period. Math. Hungar. 79 (2019), 86–93.10.1007/s10998-018-0265-9Search in Google Scholar
[20] Zhu, H.—Le, M.-H.—Soydan, G.—Togbé, A.: On the exponential Diophantine equationx2+2apb = yn, Period. Math. Hungar. 70 (2015), 233–247.10.1007/s10998-014-0073-9Search in Google Scholar
© 2022 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular Papers
- Logical and algebraic properties of generalized orthomodular posets
- Integral prefilters and integral Eq-algebras
- Modules with fusion and implication based over distributive lattices: Representation and duality
- Localization of k × j-rough Heyting algebras
- Meet infinite distributivity for congruence lattices of direct sums of algebras
- Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
- On extensions of the Open Door Lemma
- Nevanlinna theory for holomophic curves from annuli into semi-Abelian varieties
- Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions
- On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions
- Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations
- Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomials
- A Study on statistical versions of convergence of sequences of functions
- Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps
- Notes on affine Killing and two-Killing vector fields
- Prediction of the exponential fractional upper record-values
- On concomitants of generalized order statistics from generalized FGM family under a general setting
- One-parameter generalization of dual-hyperbolic Pell numbers
Articles in the same Issue
- Regular Papers
- Logical and algebraic properties of generalized orthomodular posets
- Integral prefilters and integral Eq-algebras
- Modules with fusion and implication based over distributive lattices: Representation and duality
- Localization of k × j-rough Heyting algebras
- Meet infinite distributivity for congruence lattices of direct sums of algebras
- Some exponential diophantine equations II: The equation x2 – Dy2 = kz for even k
- On extensions of the Open Door Lemma
- Nevanlinna theory for holomophic curves from annuli into semi-Abelian varieties
- Stability and feedback stabilizability of delay periodic differential equations with pairwise permutable matrix functions
- On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions
- Oscillatory and asymptotic behavior of solutions of third-order quasi-linear neutral difference equations
- Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomials
- A Study on statistical versions of convergence of sequences of functions
- Fuzzy fixed point theorems and Ulam-Hyers stability of fuzzy set-valued maps
- Notes on affine Killing and two-Killing vector fields
- Prediction of the exponential fractional upper record-values
- On concomitants of generalized order statistics from generalized FGM family under a general setting
- One-parameter generalization of dual-hyperbolic Pell numbers
