Research Article

Powers as Fibonacci sums

Published in: Quaestiones Mathematicae
Volume 48 , issue 4 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 597–609
DOI: 10.2989/16073606.2024.2411461
Author(s): Benjamin Earp-Lynch Carleton University, Canada, Simon Earp-Lynch Carleton University, Canada, Omar Kihel Brock University, Canada, P. Tiebekabe Université de Kara, Togo,
Keywords: 11B39, 11D61, 11J86, Fibonacci sequence, Diophantine equation, Baker’s method,
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Abstract

We examine the equation for positive integers y, a ≥ 2 and k ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y ≤ 25000 when k = 3, for y ≤ 1000 when k = 4, for y ≤ 40 when k = 5, and for y ≤ 3 when k = 6.
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