Abstract
Let \( \{F_{n}\}_{n\ge 0} \) be the sequence of Fibonacci numbers defined by \( F_0=0 \), \( F_1 =1\) and \( F_{n+2}= F_{n+1} +F_n\) for all \( n\ge 0 \). In this paper, for an integer \( d\ge 2 \) which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation \( x^2-dy^2=\pm 4 \) which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize.
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Acknowledgements
The first author, MD, was supported by the Austrian Science Fund (FWF), Projects F5510-N26 – Part of the Special Research Program (SFB), “Quasi-Monte Carlo Methods: Theory and Applications” and W1230 – “Doctoral Program Discrete Mathematics”. He also thanks his supervisor, Professor Robert Tichy for the encouragement and support during his Ph.D. studies in Graz. The second author, FL, was also supported by grant CPRR160325161141 from the NRF of South Africa and RTNUM18 from the CoEMaSS at Wits.
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Ddamulira, M., Luca, F. The x-coordinates of Pell equations and sums of two Fibonacci numbers II. Proc Math Sci 130, 58 (2020). https://doi.org/10.1007/s12044-020-00578-4
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DOI: https://doi.org/10.1007/s12044-020-00578-4