Abstract
In this paper, we find all Padovan and Perrin numbers which can be expressible as a products of two repdigits in the base b with \(2\le b\le 10\). It is shown that the largest Padovan and Perrin numbers which can be expressible as a products of two repdigits are \(P_{25}=616\) and \(T_{22}=486\), respectively. The proofs use lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adegbindin, C., Luca, F., Togbé, A.: Lucas numbers as sums of two repdigits. Lith. Math. J. 59, 295–304 (2019)
Adegoke, K.: Summation Identities Involving Padovan and Perrin Numbers. http://arxiv.org/abs/1812.03241v2
Batte, H., Chalebgwa, T. P., Ddamulira, M.: Perrin Numbers that are Concatenations of Two Distinct Repdigits. https://arxiv.org/abs/2105.08515
Bravo, J.J., Gómez, C.A., Luca, F.: Powers of two as sums of two k-Fibonacci numbers. Miskolc Math. Notes 17(1), 85–100 (2016)
Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas powers. Ann. Math. (2) 163, 969–1018 (2006)
Ddamulira, M.: Padovan numbers that are concatenations of two distinct repdigits. Math. Slovaca (2) 71, 275–284 (2021)
de Weger, B.M.M.: Algorithms for Diophantine Equations, CWI Tracts 65. Stichting Mathematisch Centrum, Amsterdam (1989)
Dujella, A., Pethö, A.: A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxf. Ser. (2) 49, 291–306 (1998)
Erduvan, F., Keskin, R.: Fibonacci and Lucas numbers as products of two repdigits. Turk. J. Math. 43, 2142–2153 (2019)
Erduvan, F., Keskin, R., Şiar, Z.: Repdigits base b as products of two Fibonacci numbers. Indian J. Pure Appl. Math. (2021). https://doi.org/10.1007/s13226-021-00041-8
Erduvan, F., Keskin, R., Şiar, Z.: Fibonacci or Lucas numbers which are products of two repdigits in base b. Bull. Braz. Math. Soc. New Ser. https://doi.org/10.1007/s00574-021-00243-y (2021)
Faisant, A.: On the Padovan Sequence. http://arxiv.org/abs/1905.07702v1
Luca, F.: Fibonacci and Lucas numbers with only one distinct digit. Port. Math. 57, 243–254 (2000)
Luca, F.: Repdigits as sums of three Fibonacci numbers. Math. Commun. 17, 1–11 (2012)
Marques, D., Togbé, A.: Perfect powers among Fibonomial coefficients. C. R. Acad. Sci. Paris 348, 717–720 (2010)
Sloane, N.J.A., et al.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org (2019)
Waldshmidt, M.: Diophantine Approximation on Linear Algebraic Groups?. Transcendence Properties of the Exponential Function in Several Variables. Springer, Berlin (2000)
Acknowledgements
The authors are grateful to the anonymous referees for useful comments to improve the quality of this paper. The first author is supported by IMSP, Institut de Mathématiques et de Sciences Physiques de l’Université d’Abomey-Calavi. The second author is supported by the Croatian Science Fund, Grant HRZZ-IP-2018-01-1313. The third author was supported in part by Purdue University Northwest.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Adédji, K.N., Filipin, A. & Togbé, A. Padovan and Perrin Numbers Which are Products of Two Repdigits in Base b. Results Math 76, 193 (2021). https://doi.org/10.1007/s00025-021-01502-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-021-01502-6
Keywords
- Padovan numbers
- Perrin numbers
- repdigits
- linear forms in logarithms
- Diophantine equations
- reduction method