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arXiv · 2409.02933

A Pair of Diophantine Equations Involving the Fibonacci Numbers

Xuyuan Chen·Hung Viet Chu·Fadhlannafis K. Kesumajana·Dongho Kim·Liran Li·Steven J. Miller·Junchi Yang·Chris Yao
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Abstract

Let $a, b\in \mathbb{N}$ be relatively prime. Previous work showed that exactly one of the two equations $ax + by = (a-1)(b-1)/2$ and $ax + by + 1 = (a-1)(b-1)/2$ has a nonnegative, integral solution; furthermore, the solution is unique. Let $F_n$ be the $n$th Fibonacci number. When $(a,b) = (F_n, F_{n+1})$, it is known that there is an explicit formula for the unique solution $(x,y)$. We establish formulas to compute the solution when $(a,b) = (F_n^2, F_{n+1}^2)$ and $(F_n^3, F_{n+1}^3)$, giving rise to some intriguing identities involving Fibonacci numbers. Additionally, we construct a different pair of equations that admits a unique positive (instead of nonnegative), integral solution.

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Xuyuan Chen, Hung Viet Chu, Fadhlannafis K. Kesumajana, Dongho Kim, Liran Li, Steven J. Miller, Junchi Yang, Chris Yao. 2024-08-21. A Pair of Diophantine Equations Involving the Fibonacci Numbers. https://doi.org/10.1080/00150517.2025.2455622

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