A Pair of Diophantine Equations Involving the Fibonacci Numbers

TL;DR

This paper explores solutions to specific Diophantine equations involving Fibonacci numbers, deriving explicit formulas for certain cases and discovering new identities, while also presenting a related equation with a unique positive integral solution.

Contribution

It provides explicit formulas for solutions when (a,b) are squares or cubes of Fibonacci numbers and introduces a new equation with a unique positive integral solution.

Findings

Formulas for solutions when (a,b) = (F_n^2, F_{n+1}^2) and (F_n^3, F_{n+1}^3)

New Fibonacci identities derived from these formulas

A different pair of equations with a unique positive integral solution

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.