Powers as Fibonacci Sums

Abstract

We examine the equation $y^a=\sum _{i=1}^k{F}_{n_i}$ for positive integers $y,a\ge 2$ and $k\ge 3$. 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 $y\le 25000$ when $k=3$, for $y\le 1000$ when $k=4$, for $y\le 40$ when $k=5$, and for $y\le 3$ when $k=6$.

Published in Quaestiones Mathematicae. 14 pages, 4 tables. This arXiv version contains minor corrections and includes a link to the updated SageMath code