Article

On The Diophantine equation F_n +F _m=2^a

Published in: Quaestiones Mathematicae
Volume 39 , issue 3 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 391–400

DOI: 10.2989/16073606.2015.1070377

Author(s): Jhon J. Bravo, ,, , Colombia, Florian Luca, ,, , South Africa,

Keywords: 11B39, 11J86, 11D61, Fibonacci numbers, Zeckendorf representation, linear forms in logarithms, reduction method,

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Abstract

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F _k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.

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