发表机构
Dipartimento di Matematica, Università di Trento; Dipartimento di Scienze Matematiche ”Giuseppe Luigi Lagrange”, Politecnico di Torino（特伦托大学数学系; 都灵理工学院朱塞佩·路易吉·拉格朗日数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Browkin $p$-进连分数关于二次无理数周期性的猜想，证明对任意正整数$t$，存在无穷多个整数平方根的展开周期长度为$2t$，并给出新的周期性充分条件与周期长度界，为Lagrange定理类似物研究提供工具。
AI 中文摘要
多位作者在$p$-进数域中引入并研究了连分数，其目的是发展实数连分数经典理论的类似物。在本文中，我们关注Browkin连分数算法，关于该算法仍有许多基本问题悬而未决，特别是关于二次无理数的周期展开。在本文中，我们解决了文献[6]中遗留的一个关于周期性的猜想。作为推论，我们证明：对于每个正整数$t$，存在无穷多个整数的平方根，其Browkin连分数展开是周期的，且周期长度为$2t$。这也解决了一个源于先前关于整数平方根可能周期长度的经典结果的问题。此外，我们提供了二次无理数周期性的新的充分条件，并推导了相应周期长度的显式界。这些结果也限制了Browkin连分数展开非周期的二次无理数的可能行为，因此为研究Lagrange定理的类似物是否成立提供了新工具。最后，我们还对二次无理数的Browkin展开进行了计算研究，特别关注整数平方根的行为以及寻找可能的非周期例子。
英文摘要
Continued fractions have been introduced and studied in the field of $p$--adic numbers by several authors, with the aim of developing analogues of the classical theory of real continued fractions. In this paper, we focus on Browkin's continued fraction algorithm, for which several fundamental questions remain open, especially concerning periodic expansions of quadratic irrationals. In this paper, we solve a conjecture about periodicity left open in [6]. As a consequence, we prove that, for every positive integer $t$, there exist infinitely many square roots of integers whose Browkin continued fraction expansion is periodic with period length $2t$. This also settles a problem originating from previous classical results on the possible period lengths of square roots of integers. Moreover, we provide new sufficient conditions for the periodicity of quadratic irrationals and derive explicit bounds for the corresponding period lengths. These results also restrict the possible behaviour of a quadratic irrational whose Browkin continued fraction expansion is non-periodic, and therefore provide new tools for investigating whether an analogue of Lagrange's theorem can hold. Finally, we also provide a computational study of the Browkin expansions of quadratic irrationals, with particular attention to the behaviour of square roots of integers and to the search for possible non-periodic examples.
