Abstract
Let \((F_n)_{n\ge 0}\) be the Fibonacci sequence given by \(F_{n+2}=F_{n+1}+F_n\) for \(n\ge 0\), where \(F_0=0\) and \(F_1=1\). In this paper, we explicitly find all solutions of the title Diophantine equation using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Peth\(\ddot{\text {o}}\).
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Acknowledgements
The first author would like to thank Dr. Karam Deo Shankhadhar for his many valuable comments and encouragements. The first author’s research is supported by IISER Bhopal Postdoctoral Fellowship. The second author’s collaboration to this work was made during a visit to IMPA. She thanks this institution for its hospitality and excellent working conditions, and CNPq-Brazil, for providing partial support through Universal 01/2016 - 427722/2016-0 grant.
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Patel, B.K., Chaves, A.P. On the Exponential Diophantine Equation \(F_{n+1}^{x} - F_{n-1}^{x} = F_{m}\). Mediterr. J. Math. 18, 187 (2021). https://doi.org/10.1007/s00009-021-01831-4
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DOI: https://doi.org/10.1007/s00009-021-01831-4