Gaps in binary cyclotomic polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911671324246016 |
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| author | Cafure, Antonio Cesaratto, Eda |
| author_facet | Cafure, Antonio Cesaratto, Eda |
| contents | For odd prime numbers $p < q$, let $Φ_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $Φ_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $Φ_{pq}$ are described as concatenations of words arising from iterations of a circular map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaps in binary cyclotomic polynomials Cafure, Antonio Cesaratto, Eda Number Theory 11C08 11B83 68R15 For odd prime numbers $p < q$, let $Φ_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $Φ_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $Φ_{pq}$ are described as concatenations of words arising from iterations of a circular map. |
| title | Gaps in binary cyclotomic polynomials |
| topic | Number Theory 11C08 11B83 68R15 |
| url | https://arxiv.org/abs/2507.19381 |