All Liouville Numbers are Transcendental

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

Abstrakty

EN

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] 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. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.

Słowa kluczowe

EN

Liouville numbers; Diophantine approximation; transcendental numbers; Liouville’s constant

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2017
• Tom: 25
• Numer: 1
• Strony: 49-54

Daty

wydano

2017-03-28

otrzymano

2017-02-23

online

2017-05-11

Twórcy

autor

Artur Korniłowicz

• , ,

autor

Adam Naumowicz

• , ,

autor

Adam Grabowski

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Identyfikatory

DOI

10.1515/forma-2017-0004