Periodic representations and rational approximations of square roots

TL;DR

This paper explores the use of Rédéi rational functions to derive rational approximations for square roots, unifying real and p-adic approaches through periodic continued fractions and power matrices.

Contribution

It introduces Rédéi rational functions as a tool for deriving rational approximations and connects them to periodic continued fractions in both real and p-adic contexts.

Findings

Rédéi rational functions can generate rational approximations for square roots.

Newton and Padé approximations are special cases of the derived methods.

The approach unifies real and p-adic square root approximations.

Abstract

In this paper the properties of R\'edei rational functions are used to derive rational approximations for square roots and both Newton and Pad\'e approximations are given as particular cases. As a consequence, such approximations can be derived directly by power matrices. Moreover, R\'edei rational functions are introduced as convergents of particular periodic continued fractions and are applied for approximating square roots in the field of p-adic numbers and to study periodic representations. Using the results over the real numbers, we show how to construct periodic continued fractions and approximations of square roots which are simultaneously valid in the real and in the p-adic field.