On a Pair of Diophantine Equations
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912908200378368 |
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| author | Arachchi, Sujith Uthsara Kalansuriya Chu, Hung Viet Liu, Jiasen Luan, Qitong Marasinghe, Rukshan Miller, Steven J. |
| author_facet | Arachchi, Sujith Uthsara Kalansuriya Chu, Hung Viet Liu, Jiasen Luan, Qitong Marasinghe, Rukshan Miller, Steven J. |
| contents | For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04488 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Pair of Diophantine Equations Arachchi, Sujith Uthsara Kalansuriya Chu, Hung Viet Liu, Jiasen Luan, Qitong Marasinghe, Rukshan Miller, Steven J. Number Theory 11D04, 11B83 For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases. |
| title | On a Pair of Diophantine Equations |
| topic | Number Theory 11D04, 11B83 |
| url | https://arxiv.org/abs/2309.04488 |