On X-coordinates of Pell equations which are repdigits

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 \(b\in [2,100]\).

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Correspondence to Carlos A. Gómez.

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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.

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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

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Keywords

  • Repdigits
  • Linear forms in logarithms
  • Baker’s method

Mathematics Subject Classification

  • 11B39
  • 11J86