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On algebraic independence of three p-adic continued fractions

Sarra Ahallal, Mohamed Begare, Ali Kacha

Abstract

In this paper, we establish sufficient conditions on the elements of the p-adic continued fractions $A$ and $B$ which guarantee that the p-adic continued fractions $A, B $ and $A^{B}$ are algebraically independent over $\mathbb{Q}$. These elements have partial quotients that increase rapidly. We note that these results extend some work of Bundschuh. Furthermore, we give some numerical examples which illustrated the theoretical results.

On algebraic independence of three p-adic continued fractions

Abstract

In this paper, we establish sufficient conditions on the elements of the p-adic continued fractions and which guarantee that the p-adic continued fractions and are algebraically independent over . These elements have partial quotients that increase rapidly. We note that these results extend some work of Bundschuh. Furthermore, we give some numerical examples which illustrated the theoretical results.

Paper Structure

This paper contains 7 sections, 1 theorem, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and auxiliary results
  3. Définition of a p-adic continued fraction
  4. Convergents of a p-adic continued fraction
  5. Main result
  6. The p-adic continued fraction of $A, B$ and $A^B$
  7. Numerical application

Key Result

Theorem 3.1

Let $A\in (1+p\mathbb{Z}_p), B\in (p\mathbb{Z}_p)$ and $(\alpha)$ be a real number $>2.$ If then the p-adic continued fractions $A,B$ and $A^B$ are algebraically independent over $\mathbb{Q}.$$\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (1)

  • Theorem 3.1