Periodic Representations and Approximations of p-adic Numbers Via Continued Fractions

Abstract

Continued fractions can be introduced in the field of p-adic numbers ${\mathbb{Q}}_p$, however currently there is not a standard algorithm as in $\mathbb{R}$. Indeed, it is not known how to construct p-adic continued fractions that give periodic representations for all quadratic irrationals and provide good p-adic approximations. In this article, we introduce a novel algorithm which terminates in a finite number of steps when processes rational numbers. Moreover, we study when it provides particular periodic representations of period 2 and pre-period 1 for quadratic irrationals. We also provide some numerical experiments regarding periodic representations and p-adic approximations of quadratic irrationals, comparing the performances with Browkin’s algorithm presented in [6], which is one of the most classical and interesting algorithm for continued fractions in ${\mathbb{Q}}_p$.

Keywords:

• Continued fractions
• p-adic approximations
• p-adic numbers
• quadratic irrationals

MSC 2010:

• 11J70
• 12J25

Declaration of interest

No potential conflict of interest was reported by the author(s).