论文标题
$ \ sqrt {m} $的某些定期整数持续分数扩展和pell方程的应用
Some periodic integer continued fraction expansions of $\sqrt{m}$ and application to the Pell equations
论文作者
论文摘要
周期性整数持续分数（PICF）是常规周期性持续分数（RPCF）的概括。经典的RPCF扩展非理性数字是唯一的。但是，对于PICF扩展而言，它不再是唯一的。因此，确定所有非理性数字扩展是一个自然的问题。在本文中，我们确定了无正方形整数的平方根的某些类型的PICF膨胀。为了获得此结果，它在Brock-Elkies-Jordan中出现的某些PCF品种的整数点起着重要作用。作为这些结果的应用，我们从无正方形整数的平方根以及RPCF扩展的PICF扩展中获得了Pell方程的基本解决方案。
Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF expansion. Hence it is a natural problem to determine all PICF expansions of irrational numbers. In this paper, we determine certain type PICF expansions of square roots of positive square-free integers. To obtain this result, it plays an important role to determine integer points on certain PCF varieties appeared in Brock-Elkies-Jordan. As an application of these results, we obtain fundamental solutions of the Pell equations from PICF expansions of square roots of positive square-free integers as well as the RPCF expansions.
