A Pair of Diophantine Equations Involving the Fibonacci Numbers

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.