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A Pair of Diophantine Equations Involving the Fibonacci Numbers by Chen, Xuyuan; Chu, Hung Viet; Kesumajana, Fadhlannafis K.; Kim, Dongho; Li, Liran; Miller, Steven J.; Yang, Junchi; Yao, Chris is a scholarly article available to read on EtoBox.

What is A Pair of Diophantine Equations Involving the Fibonacci Numbers about?

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.

Author

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

Published

2024

Language

EN