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

Abstract

Continued fractions can be introduced in the field of p-adic numbers Q p , however currently there is not a standard algorithm as in 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 Q p .