Introduction to Liouville Numbers

• Autorzy: Adam Grabowski , Artur Korniłowicz
• Języki publikacji: EN

Abstrakty

EN

The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and [...] It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum [...] for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as [...] substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https://en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.

Słowa kluczowe

EN

Liouville number; Diophantine approximation; transcendental number; Liouville constant

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2017
• Tom: 25
• Numer: 1
• Strony: 39-48

Daty

wydano

2017-03-28

otrzymano

2017-02-23

online

2017-05-11

Twórcy

autor

Adam Grabowski

• , ,

autor

Artur Korniłowicz

• , ,

Identyfikatory

DOI

10.1515/forma-2017-0003