On Pillai’s problem with the Fibonacci and Pell sequences

Hernández Hernández, Santos; Luca, Florian; Rivera, Luis Manuel

doi:10.1007/s40590-018-0223-9

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On Pillai’s problem with the Fibonacci and Pell sequences

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Luca, Florian
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s40590-018-0223-9
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Hernandez-Luca-Rivera_On Pillai’s problem with the Fibonacci and Pell sequences_Preprint.pdf
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Citation

Hernández Hernández, S., Luca, F., & Rivera, L. M. (2019). On Pillai’s problem with the Fibonacci and Pell sequences. Boletín de la Sociedad Matemática Mexicana, 25(3), 495-507. doi:10.1007/s40590-018-0223-9.

Cite as: https://hdl.handle.net/21.11116/0000-0004-ED02-F

Abstract

Let (Fn)n≥0 and (Pn)n≥0 be the Fibonacci and Pell sequences given by the initial conditions F0=0, F1=1, P0=0, P1=1 and the recurrence formulas Fn+2=Fn+1+Fn, Pn+2=2Pn+1+Pn for all n≥0, respectively. In this note, we study the Pillai type problem:
Fn−Pm=Fn1−Pm1
in non-negative integer pairs (n,m)≠(n1,m1). We completely solve this equation.