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On Pillai’s problem with the Fibonacci and Pell sequences

Abstract

Let \((F_n)_{n\ge 0}\) and \((P_n)_{n\ge 0}\) be the Fibonacci and Pell sequences given by the initial conditions \(F_0=0,~F_1=1\), \(P_0=0, ~P_1=1\) and the recurrence formulas \(F_{n+2}=F_{n+1}+F_n\),  \(P_{n+2}=2P_{n+1}+P_n\) for all \(n\ge 0\), respectively. In this note, we study the Pillai type problem:

$$\begin{aligned} F_n-P_m=F_{n_1}-P_{m_1} \end{aligned}$$

in non-negative integer pairs \((n,m)\ne (n_1,m_1)\). We completely solve this equation.

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Acknowledgements

We thank the referees for their comments and remarks which improved the quality of this work. S.H.H. thanks Leticia A. Ramírez for the generous support. F. L. was supported in part by Grant CPRR160325161141 and an A-rated scientist award both from the NRF of South Africa and by Grant no. 17-02804S of the Czech Granting Agency. F.L. worked on this paper during visits to the Mathematics Department at the UAZ, Zacatecas, Mexico, and the Max Planck Institute for Mathematics, Bonn, Germany, in February and March of 2018, respectively. He thanks these institutions for the generous support and hospitality. L.M.R. was partially supported by Grant PFCE 2018-2019 (UAZ-CA-169).

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Correspondence to Santos Hernández Hernández.

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Hernández Hernández, S., Luca, F. & Rivera, L.M. On Pillai’s problem with the Fibonacci and Pell sequences. Bol. Soc. Mat. Mex. 25, 495–507 (2019). https://doi.org/10.1007/s40590-018-0223-9

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Keywords

  • Fibonacci
  • Pell sequences
  • Pillai’s type problem
  • Linear form in logarithms

Mathematics Subject Classification

  • 11B39
  • 11D45
  • 11D61
  • 11J86