On Liouville Sequences in the Non-Archimedean Case

Hamza Menken,Abdulkadir Aşan

doi:10.1155/2014/813924

Abstract

We study Liouville numbers in the non-Archimedean case. We deal with the concept of a Liouville sequence in the non-Archimedean case and we give some results both in thep-adic numbers fieldℚpand the functions fieldK〈x〉.

Highlights

It is well known that if a complex number α is a root of a nonzero polynomial equation anxn + an−1xn−1 + ⋅ ⋅ ⋅ + a1x + a0 = 0, (1)
Where the ais are integers and α satisfies no similar equation of degree < n, α is said to be an algebraic number of degree n
The classical Liouville numbers are real numbers that can be rapidly approximated by rational numbers, the p-adic Liouville numbers are those numbers that can be rapidly approximated by positive integers in the p-adic norm

Summary

Introduction

Liouville’s theorem can be applied to prove the transcendence of a large class of real numbers which are called Liouville numbers. The natural numbers set N is dense in Zp. the classical Liouville numbers are real numbers that can be rapidly approximated by rational numbers, the p-adic Liouville numbers are those numbers that can be rapidly approximated by positive integers in the p-adic norm. The p-adic Liouville numbers are defined as follows. Λ is called a p-adic Liouville number. According to this definition, λ ∈ Zp is a p-adic Liouville number if and only if there exists a sequence of positive integers an such that nl→im∞√n 󵄨󵄨󵄨󵄨an − λ󵄨󵄨󵄨󵄨p = 0. Liouville numbers have been studied in [17,18,19,20,21] and others

Liouville Sequences in the p-adic Numbers Fields

The Liouville Sequences in the Functions Field