On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
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Florian Luca
Abstract
In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence.
Acknowledgement
The authors thank the referees for the useful comments to improve the quality of this paper.
(Communicated by István Gaál
References
[1] Baker, A.—Davenport, H.: The equations 3x2 − 2 = y2 and 8x2 − 7 = z2, Q. J. Math. 20(1) (1969), 129–137.10.1093/qmath/20.1.129Search in Google Scholar
[2] Behera, A.—Liptai, K.—Panda, G. K.—Szalay, L.: Balancing with Fibonacci powers, Fibonacci Quart. 49 (2011), 28–33.Search in Google Scholar
[3] Bertók, C.—Hajdu, L.—Pink, I.—Rábai, Z.: Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory 13(2) (2017), 4–5.10.1142/S1793042117500166Search in Google Scholar
[4] Bravo, J. J.—Luca, F.: Coincidences in generalized Fibonacci recurrences, J. Number Theory 133(6) (2013), 2121–2137.10.1016/j.jnt.2012.11.006Search in Google Scholar
[5] Carmichael, R. D.: On the numerical factors of the arithmetic forms αn ± βn, Ann. Math. (2) 15(2) (1913), 30–70.10.2307/1967797Search in Google Scholar
[6] Dujella, A. and Pethő, A.: A generalization of a theorem of Baker and Davenport, Q. J. Math. Oxford Ser. 49(3) (1998), 291–306.10.1093/qmathj/49.3.291Search in Google Scholar
[7] Laurent, M.—Mignotte, M.—Nesterenko, Yu.: Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55(2) (1995), 285–321.10.1006/jnth.1995.1141Search in Google Scholar
[8] Luca, F.—Oyono, R.: Exponential Diophantine equation related to powers of two consecutive Fibonacci numbers, Proc. Japan Acad. 87 (2011), 45–50.10.3792/pjaa.87.45Search in Google Scholar
[9] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Math. 64(6) (2000), 1217–1269.10.1070/IM2000v064n06ABEH000314Search in Google Scholar
[10] Murty, M. R.—Esmonde, J.: Problems in Algebraic Number Theory. 2nd ed., Grad. Texts in Math. 190, Springer-Verlag, New York, 2005.10.1007/978-3-642-87939-5_1Search in Google Scholar
[11] Pethö, A.: The Pell sequence contains only trivial perfects powers: Sets, graphs and numbers, Colloq. Math. Soc. Janós Bolyai 60 (1991), 561–568.Search in Google Scholar
[12] Rihane, S. E.—Faye, B.—Luca, F.—Togbé, A.: On the exponential Diophantine equation
[13] Soydan, G.—Németh, L.—Szalay, L.: On the Diophantine equation
© 2020 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- Fuzzy deductive systems of RM algebras
- Congruence pairs of principal MS-algebras and perfect extensions
- The lattices of 𝔏-fuzzy state filters in state residuated lattices
- Central lifting property for orthomodular lattices
- EQ-Modules
- On the exponential Diophantine equation Pxn + Pxn+1 + ⋯ + Pxn+k-1 = Pm
- Remarks on some generalization of the notion of microscopic sets
- Disjointness of composition operators on Hv0 spaces
- A common fixed point theorem for non-self mappings in strictly convex menger PM-spaces
- The Poincaré-Cartan forms of one-dimensional variational integrals
- Coarse cohomology with twisted coefficients
- Divisible extension of probability
- Asymptotic behavior of the records of multivariate random sequences in a norm sense
- Strong convergence of the functional nonparametric relative error regression estimator under right censoring
- A new kumaraswamy generalized family of distributions: Properties and applications
- Efficient message transmission via twisted Edwards curves
- Computation of several Hessenberg determinants
