Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917537594212352 |
|---|---|
| author | Miller, Steven J. Nikolakopoulos, Dimitrios Srinivasan, Anitha |
| author_facet | Miller, Steven J. Nikolakopoulos, Dimitrios Srinivasan, Anitha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19083 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation Miller, Steven J. Nikolakopoulos, Dimitrios Srinivasan, Anitha Number Theory 11D68, 11D72 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. |
| title | Fibonacci Numbers and Vieta Jumping for a Rational Diophantine Equation |
| topic | Number Theory 11D68, 11D72 |
| url | https://arxiv.org/abs/2605.19083 |