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Mathematics Stack Exchange

Questions tagged [transcendence-theory]

A brief study of transcendental numbers and algebraic independence theories; currently an on-development branch of mathematics with a lot of open problems.

Primarily, questions related to determination of transcendental numbers, theory of algebraic independence and diophantine approximation should have this tag. Closed form-related question can have this tag if is broad enough. Example: is $\text{K} \cap \mathbb{Q} = \lbrace\emptyset\rbrace$ for the ring $\text{K} \in \mathbb{C}$ defined as $(...)$?

Also known as transcendental number theory.

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How to show $e^{e^{e^{79}}}$ is not an integer

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? After several years, it appears this is an open problem. As a non-number…
Carl Mummert
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Is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Let $$x=0.112123123412345123456\dots $$ Since the decimal expansion of $x$ is non-terminating and non-repeating, clearly $x$ is an irrational number. Can it be shown whether $x$ is algebraic or transcendental over $\mathbb{Q}$ ? I think $x$ is…
ASB
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Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
user110661
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Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
Vladimir Reshetnikov
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Values of hypergeometric functions

Let $_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;c)$ denote the generalized hypergeometric function. Let $A \subset \mathbb R$ be the set of all values of $\ _pF_q(\cdot)$ at rational points $a_i,b_j,c\in \mathbb Q$. Since $A$ and its algebraic closure…
Igor Pak
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Fields the closure of which is $\mathbb{C}$

Studying Galois Theory I have finally done the Fundamental Theorem of Algebra, which simply states that $\overline{\mathbb{R}} = \mathbb{C}$. My question is: do there exist other fields different from $\mathbb{R}, \mathbb{C}$ such that their…
JayTuma
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Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be either algebraic or transcendental?
Charles
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Irrational numbers, decimal representation

Can this even be proved? (Or disproved?) Any irrational number without a 0 (zero) in its decimal representation is transcendental. Not sure where to start on this one...
ManRow
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Existence of an entire function with algebraically independent derivatives

Let $\mathbb{A}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. A collection of functions $F=\lbrace f_i:X \rightarrow\mathbb{C}\rbrace$ is said to be algebraically independent over $\mathbb{Q}$ at $x \in X$ if the $f_i(x)$ are distinct,…
Logan M
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Is $\Bbb Q(\sqrt 2, e)$ a simple extension of $\Bbb Q$?

My general question is to find, if this is possible, two real numbers $a,b$ such that $K=\Bbb Q(a,b)$ is not a simple extension of $\Bbb Q$. $\newcommand{\Q}{\Bbb Q}$ Of course $a$ and $b$ can't be both algebraic, otherwise $K$ would be a…
Watson
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Irrationality measure of the Chaitin's constant $\Omega$

What is known about irrationality measure of the Chaitin's constant $\Omega$? Is it finite? Can it be a computable number? Can it be $2$?
Vladimir Reshetnikov
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Integer solutions of $\log a \log b = \log c \log d$

Four positive integers $a,b,c,d>1$ satisfy $\log a \log b = \log c \log d$. Is necessarily $\frac{\log a}{\log c} \in \mathbb{Q}$ or $\frac{\log a}{\log d} \in \mathbb{Q}$? I tried to use Gelfond–Schneider theorem and Lindemann–Weierstrass theorem…
Myeung Su Kim
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Is the sum of the inverses of the factorials of Fibonacci numbers transcendental?

This is the sum, $$e'=\sum_{n=1}^\infty \frac{1}{F_{n}!}$$ where $F_{n}$ is the $n^{th}$ Fibonacci number. Is it possible to prove that it will converge to a transcendental number? Edit: Proof of irrationality-: But first some lemmas, $F_q\geq q…
LMS
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What methods show that a number is transcendental?

I've been doing a lot of research on such theories lately and these are all I've found so far: Liouvilles criterion (here) Lindemann-Weierstrass theorem (here) Gelfond-Schneider theorem (here) Brownawell-Waldschmidt theorem (here or here) Schanuels…
Kinheadpump
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Is the number transcendental?

Consider the following number: $R=\frac{1}{9}\sum^\infty_{n=1}…
El Ectric
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