Abstract
In this paper, we give an algorithm which finds, for an integer base \(b\ge 2\), all squarefree integers \(d\ge 2\) such that sequence of X-components \(\{X_n\}_{n\ge 1}\) of the Pell equation \(X^2-dY^2=\pm 1\) has two members which are base b-repdigits. We implement this algorithm and find all the solutions to this problem for all bases \(b\in [2,100]\).
Similar content being viewed by others
References
Baker, A.: On the representation of integers by binary forms. Philos. Trans. A 263, 173–208 (1968)
Baker, A., Wüstholz, G.: Logarithmic Forms and Diophantine Geometry, vol. 9. Cambridge University Press, Cambridge (2008)
Bennett, M.A., Pintér, A.: Intersections of recurrence sequences. Proc. Am. Math. Soc. 143, 2347–2353 (2015)
Bravo, J.J., Luca, F.: On a conjecture about repdigits in \(k\)-generalized Fibonacci sequences. Publ. Math. Debrecen 82, 623–639 (2013)
Bugeaud, Y., Mignotte, M.: On integers with identical digits. Mathematika 46, 411–417 (1999)
Cohen, H.: Number Theory. Tools and Diophantine Equations, vol. 1. Springer, New York (2007)
Dossavi-Yovo, A., Luca, F., Togbé, A.: On the \(x\)-coordinates of Pell equations which are rep-digits. Publ. Math. Debrecen 88, 381–399 (2016)
Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. 49, 291–306 (1998)
Faye, B., Luca, F.: On \(X\)-coordinates of Pell equations that are repdigits. Fibonacci Q. 56, 52–62 (2018)
Guzmán-Sanchez, S., Luca, F.: Linear combinations of factorials and S-units in a binary recurrence sequence. Ann. Math. Québec 38, 169–188 (2014)
Luca, F., Togbé, A.: On the \(x\)-coordinates of Pell equations which are Fibonacci numbers. Math. Scand. 122, 18–30 (2018)
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), in Russian; English translation in Izv. Math. 64, 1217–1269 (2000)
Murty, R.M., Esmonde, J.: Problems in Algebraic Number Theory. Graduate Texts in Mathematics, vol. 190, 2nd edn. Springer, New York (2005)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Acknowledgements
We thank the referee for comments which improved the quality of our paper. C. A. G. was supported in part by Project 71228 (Universidad del Valle). F. L. was supported by grant RTNUM20 from CoEMaSS, Wits, South Africa. F. S. Z. was supported by a Ph.D. grant from the NRF of South Africa.
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
Gómez, C.A., Luca, F. & Zottor, F.S. On X-coordinates of Pell equations which are repdigits. Res. number theory 6, 41 (2020). https://doi.org/10.1007/s40993-020-00220-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-020-00220-2