Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

We study the Diophantine equation ${\displaystyle \frac{a+1}{b}+\frac{b+1}{a}=k}$, where $k$ is an integer. Using Vieta jumping, we completely classify all positive integer pairs $(a,b)$. We prove that the associated integer value $k$ can only be $3$ or $4$. The corresponding solution pairs $(a,b)$ are related to the classical Fibonacci numbers. As a consequence, the quantity $\frac{a+b}{\gcd (a,b{)}^2}$ takes only the values $1,2,3$ and $5$. This reveals an unexpected connection between a simple rational Diophantine condition, Vieta jumping, and Fibonacci numbers.