Research Article

Powers as Fibonacci sums

Published in: Quaestiones Mathematicae
Volume 48 , issue 4 , 2025 , 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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