On rational approximations of algebraic numbers of higher orders and certain parametrization for generalized Pell's equations

TL;DR

This paper introduces recurrent fractions, a new algebraic structure generalizing continued fractions, and uses them to develop an algorithm for approximating algebraic irrationals and to parametrize generalized Pell's equations.

Contribution

It presents a novel algebraic object called recurrent fractions and applies it to rational approximation algorithms and Pell's equations parametrization.

Findings

Recurrent fractions generalize continued fractions to n-dimensions.

An algorithm for rational approximation of algebraic irrationals is proposed.

A parametrization for generalized Pell's equations is constructed.

Abstract

A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.