Abstract
For an integer \( k \ge 2 \), let \( \{F^{(k)}_{n} \}_{n\ge 0}\) be the k–generalized Fibonacci sequence which starts with \( 0, \ldots , 0, 1 \) (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed \(k \ge 4\). This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.1007/s12044-017-0338-3) for the case \(k=2\) and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486) for the case \(k=3\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, A., Davenport, H.: The equations \(3x^2-2 =y^2\) and \(8x^2-7=z^2\). Quart. J. Math. Oxford Ser. 20(2), 129–137 (1969)
Baker, A., Wüstholz, G.: Logarithmic Forms and Diophantine Geometry, Volume 9 of New Mathematical Monographs. Cambridge University Press, Cambridge (2007)
Bravo, J.J., Luca, F.: Powers of two in generalized Fibonacci sequences. Revista Colombiana de Matemáticas 46(1), 67–79 (2012)
Mihăilescu, P. P.: Primary cyclotomic units and a proof of Catalans conjecture Journal für die reine und angewandte Mathematik (Crelles Journal), 2004. 572 (2006): 167–195. Retrieved 24 July 2017, https://doi.org/10.1515/crll.2004.048
Bravo, J.J., Luca, F., Yazán, K.: On a problem of Pillai with Tribonacci numbers and powers of 2. Bull. Korean Math. Soc. 54(3), 1069–1080 (2017). https://doi.org/10.4134/BKMS.b160486
Chim, K.C., Pink, I., Ziegler, V.: On a variant of Pillai’s problem. Int. J. Number Theory 13(7), 1711–1727 (2017). https://doi.org/10.1142/S1793042117500981
Cooper, C., Howard, F.T.: Some identities for \(r-\)Fibonacci numbers. Fibonacci Quart. 49(3), 231–243 (2011)
Ddamulira, M., Luca, F., Rakotomalala, M.: On a problem of Pillai with Fibonacci numbers and powers of 2. Proc. Math. Sci. 127(3), 411–421 (2017). https://doi.org/10.1007/s12044-017-0338-3
Dresden, G. P., Du, Zhaohui: A simplified Binet formula for \(k-\)generalized Fibonacci numbers. J. Integer Seq. 17, Article 14.4.7 (2014)
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. 49(3), 291–306 (1998)
Gómez Ruíz, C.A., Luca, F.: On the largest prime factor of the ratio of two generalized Fibonacci numbers. J. Number Theory 152, 182–203 (2015)
Herschfeld, A.: The equation \(2^x - 3^y = d\). Bull. Amer. Math. Soc. 41, 631 (1935)
Herschfeld, A.: The equation \(2^x - 3^y = d\). Bull. Amer. Math. Soc. 42, 231–234 (1936)
Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64(6):125–180 (2000); translation in Izv. Math. 64 , No. 6, pp. 1217–1269 (2000)
Pillai, S.S.: On \(a^x - b^y = c\). J. Indian Math. Soc. 2, 119–122 (1936)
Pillai, S.S.: A correction to the paper On \(a^x - b^y = c\). J. Indian Math. Soc. 2, 215 (1937)
Acknowledgements
We thank the referee for pointing out some errors in a previous version of this manuscript. MD was supported by the FWF Grants: F5510-N26 which is part of the special research program (SFB), “Quasi-Monte Carlo Methods: Theory and Applications”, P26114-N26 with the title “Diophantine Problems” and W1230 which is the “Doctoral Program Discrete Mathematics”. Part of the work was done when MD visited the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Vienna. He would like to thank this institution for its hospitality and a fruitful working environment.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ddamulira, M., Gómez, C.A. & Luca, F. On a problem of Pillai with k–generalized Fibonacci numbers and powers of 2. Monatsh Math 187, 635–664 (2018). https://doi.org/10.1007/s00605-018-1155-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-018-1155-1