Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911909786157056 |
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| author | Barbero, Stefano Cerruti, Umberto Murru, Nadir Salvatori, Giulia |
| author_facet | Barbero, Stefano Cerruti, Umberto Murru, Nadir Salvatori, Giulia |
| contents | We present a characterization of the algebraic integers with continued fraction expansions of the form $[a_0, \overline{a_1, \ldots, a_n, s}]$, where $(a_1, \ldots, a_n)$ is a palindrome and $s \in \mathbb{N}_{\geq 1}$. In particular, we focus on the special case where $(a_1, \ldots, a_n) = (m, \ldots, m)$, providing a detailed characterizations of the corresponding algebraic integers and $s$ in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given $m$ and $n$. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell's equations corresponding to this family of integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06430 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods Barbero, Stefano Cerruti, Umberto Murru, Nadir Salvatori, Giulia Number Theory We present a characterization of the algebraic integers with continued fraction expansions of the form $[a_0, \overline{a_1, \ldots, a_n, s}]$, where $(a_1, \ldots, a_n)$ is a palindrome and $s \in \mathbb{N}_{\geq 1}$. In particular, we focus on the special case where $(a_1, \ldots, a_n) = (m, \ldots, m)$, providing a detailed characterizations of the corresponding algebraic integers and $s$ in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given $m$ and $n$. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell's equations corresponding to this family of integers. |
| title | Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.06430 |