scholarly journals All Liouville Numbers are Transcendental

2017 ◽  
Vol 25 (1) ◽  
pp. 49-54
Author(s):  
Artur Korniłowicz ◽  
Adam Naumowicz ◽  
Adam Grabowski
Keyword(s):  
Positive Integer ◽  
Diophantine Approximation ◽  
Algebraic Number ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Liouville Number ◽  
Liouville Numbers ◽  
Transcendental Numbers ◽  
Algebraic Domains ◽  
Liouville’S Theorem

Summary 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.

scholarly journals Introduction to Liouville Numbers

2017 ◽  
Vol 25 (1) ◽  
pp. 39-48
Author(s):  
Adam Grabowski ◽  
Artur Korniłowicz
Keyword(s):  
Positive Integer ◽  
Algebraic Number ◽  
Finite Sequence ◽  
Rational Numbers ◽  
Liouville Number ◽  
Liouville Numbers ◽  
Constant Sequence ◽  
Transcendental Numbers ◽  
Finite Sequences ◽  
Definition Of

Summary 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.

scholarly journals The Completed Finite Period Map and Galois Theory of Supercongruences

2018 ◽  
Vol 2019 (23) ◽  
pp. 7379-7405
Author(s):  
Julian Rosen
Keyword(s):  
Algebraic Number ◽  
Complex Number ◽  
Galois Theory ◽  
Rational Numbers ◽  
Multiple Zeta Values ◽  
Period Map ◽  
Transcendental Numbers ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Finite Period

Abstract A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals Algebraic values of sines and cosines and their arguments

2021 ◽  
Vol 61 ◽  
pp. 21-28
Author(s):  
Edmundas Mazėtis ◽  
Grigorijus Melničenko
Keyword(s):  
Rational Numbers ◽  
Trigonometric Functions ◽  
Algebraic Numbers ◽  
Transcendental Numbers

The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these functions are transcendental numbers. Hence, it follows that the values of all angles of the pseudo-Heronian triangle, including the values of all angles of the Pythagoras or Heron triangle, expressed in radians, are transcendental numbers. If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. It should be noted that in this case the argument x = r 2\pi  is transcendental and, if expressed in degrees, becomes a rational.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Multiple Legendre polynomials in diophantine approximation

2014 ◽  
Vol 10 (07) ◽  
pp. 1829-1855 ◽  
Author(s):  
Raffaele Marcovecchio
Keyword(s):  
Recent Result ◽  
Diophantine Approximation ◽  
Legendre Polynomials ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Algebraic Numbers ◽  
Bounded Degree

We construct a class of multiple Legendre polynomials and prove that they satisfy an Apéry-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. We prove, e.g., that the nonquadraticity exponent of log 2 is bounded from above by 12.841618…, thus improving upon a recent result of the author. Our construction also yields some other known results.

scholarly journals On the approximation of algebraic numbers by algebraic integers

1963 ◽  
Vol 3 (4) ◽  
pp. 408-434 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Positive Constant ◽  
Algebraic Number Field ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Image Position ◽  
Rational Integral

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

scholarly journals Algebraic Numbers

2016 ◽  
Vol 24 (4) ◽  
pp. 291-299
Author(s):  
Yasushige Watase
Keyword(s):  
Polynomial Ring ◽  
Algebraic Number ◽  
Algebraic Integer ◽  
Commutative Rings ◽  
Formal Proof ◽  
Integral Closure ◽  
Ring Extension ◽  
Algebraic Numbers ◽  
Transcendental Numbers ◽  
Integral Element

Summary This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.

On the approximation of quadratic algebraic numbers

2005 ◽  
Vol 42 (2) ◽  
pp. 195-205
Author(s):  
Sándor H.-Molnár
Keyword(s):  
Positive Integer ◽  
Algebraic Number ◽  
Algebraic Integer ◽  
Second Order ◽  
Algebraic Numbers ◽  
Order Linear ◽  
Recursive Sequences ◽  
Approximating Sequence ◽  
Real Quadratic

In the paper we construct such second order linear recursive sequences G and H of rational integers that with their terms |a -Gn+1 /H n| < 1/ (\sqrtvDH2n) holds for every positive integer n, where a denotes a real quadratic algebraic integer of discriminant D. An approximating sequence of the form Gn+1 /Hn is also given for a  if it is only a real quadratic algebraic number (not an algebraic integer), but in this case the approximating constant is not the best.

Sets with large intersection and ubiquity

2008 ◽  
Vol 144 (1) ◽  
pp. 119-144 ◽  
Author(s):  
ARNAUD DURAND
Keyword(s):  
Diophantine Approximation ◽  
Nondecreasing Function ◽  
Gauge Function ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Integer Polynomials ◽  
Large Intersection ◽  
Image Position ◽  
Full Lebesgue Measure ◽  
Arbitrary Norm

AbstractA central problem motivated by Diophantine approximation is to determine the size properties of subsets of$\R^d$ ($d\in\N$)of the formwhere ‖⋅‖ denotes an arbitrary norm,Ia denumerable set, (xi,ri)i∈ Ia family of elements of$\R^d\$× (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that ifFId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subsetVof$\R^d\$, the setFϕbelongs to a class Gh(V) of sets with large intersection inVwith respect to a given gauge functionh. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorffg-measure for every gauge functiongwhich increases faster thanhnear zero. In particular, this yields a sufficient condition on a gauge functiongsuch that a given countable intersection of sets of the formFϕhas infinite Hausdorffg-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequenceψof positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that areψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.

Sign in / Sign up

Export Citation Format

Share Document