A Pair of Diophantine Equations Involving the Fibonacci Numbers
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| Format: | Preprint |
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2024
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| _version_ | 1866909305745178624 |
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| author | Chen, Xuyuan Chu, Hung Viet Kesumajana, Fadhlannafis K. Kim, Dongho Li, Liran Miller, Steven J. Yang, Junchi Yao, Chris |
| author_facet | Chen, Xuyuan Chu, Hung Viet Kesumajana, Fadhlannafis K. Kim, Dongho Li, Liran Miller, Steven J. Yang, Junchi Yao, Chris |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_02933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Pair of Diophantine Equations Involving the Fibonacci Numbers Chen, Xuyuan Chu, Hung Viet Kesumajana, Fadhlannafis K. Kim, Dongho Li, Liran Miller, Steven J. Yang, Junchi Yao, Chris Number Theory 11B39, 11D04 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. |
| title | A Pair of Diophantine Equations Involving the Fibonacci Numbers |
| topic | Number Theory 11B39, 11D04 |
| url | https://arxiv.org/abs/2409.02933 |