Convergence, finiteness and periodicity of several new algorithms of $p$-adic continued fractions

Abstract:

Classical continued fractions can be introduced in the field of $p$-adic numbers, where $p$-adic continued fractions offer novel perspectives on number representation and approximation. While numerous $p$-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in $\mathbb {Q}_p$ for a given prime $p$. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes $p$, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes $p$, we propose a potential approach to establish a $p$-adic continued fraction expansion algorithm. As before, the algorithm is designed to expand algebraic numbers in $\mathbb {Q}_p$, while generating periodic expansions for all quadratic irrationals in $\mathbb {Q}_p$.