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arXiv · 2608.04445

Powers as Fibonacci Sums

Benjamin Earp-Lynch·Simon Earp-Lynch·Omar Kihel·Pagdame Tiebekabe
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Abstract

We examine the equation $y^{a}=\sum\limits_{i=1}^{k}F_{n_{i}}$ for positive integers $y,a\geq 2$ and $k\geq3$. 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\leq 25000$ when $k=3$, for $y\leq 1000$ when $k= 4$, for $y\leq 40$ when $k= 5$, and for $y\leq 3$ when $k=6$.

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Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel, Pagdame Tiebekabe. 2026-08-05. Powers as Fibonacci Sums. https://doi.org/10.2989/16073606.2024.2411461

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