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Linear independence of continued fractions with algebraic terms

Jaroslav Hančl, Mathias L. Laursen, Jitu Berhanu Leta

Abstract

We give conditions on sequences of positive algebraic numbers $\{a_{n,j}\}_{n=1}^\infty$, $j=1,\dots ,M$ and number field $\mathbb K$ to ensure that the numbers defined by the continued fractions $[0;a_{1,j},a_{2,j},\dots ]$, $j=1,\dots ,M$ and $1$ are linearly independent over $\mathbb K$.

Linear independence of continued fractions with algebraic terms

Abstract

We give conditions on sequences of positive algebraic numbers , and number field to ensure that the numbers defined by the continued fractions , and are linearly independent over .

Paper Structure

This paper contains 4 sections, 10 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main Results
  4. Proofs

Key Result

Theorem 1

Let $\{a_n\}_{n=1}^\infty$ be a non-decreasing sequence of positive integers such that $\lim\limits_{n\rightarrow\infty} a_n^{\frac{1}{3^n}} = \infty$ and $\{ p_n\}_{n=1}^\infty$ be the increasing sequence of all primes. Then the continued fractions and number $1$ are linearly independent over $\mathbb Q(\sqrt 2)$ particularly over $\mathbb Q$ .

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Example 5
  • Corollary 6
  • Example 7
  • Example 8
  • Corollary 9
  • Corollary 10
  • ...and 18 more