A Pair of Diophantine Equations Involving the Fibonacci Numbers

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.

12 pages