Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation

TL;DR

This paper classifies solutions to a specific rational Diophantine equation using Vieta jumping, revealing connections to Fibonacci numbers and showing that the parameter k can only be 3 or 4.

Contribution

It completely characterizes positive integer solutions to the equation, linking them to Fibonacci numbers and identifying the limited possible values of k.

Findings

k can only be 3 or 4

Solution pairs relate to Fibonacci numbers

The ratio (a+b)/gcd(a,b)^2 takes values 1, 2, 3, 5

Abstract

We study the Diophantine equation $\displaystyle{\tfrac{a+1}{b} + \tfrac{b+1}{a} \ = \ k}$, 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.