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

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$V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. $V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$, where $k(V)$ ...
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2answers
55 views

For what values of $x$ is $\cos x$ transcendental?

For what values of $x$ is $\cos x$ transcendental? Is there any way I can figure out the values of $x$ where $\cos x$ is transcendental or do I have to check individually for every $x$ whether it is ...
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1answer
29 views

Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} ...
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25 views

Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
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27 views

Constants in Siegel's Lemma

I've got a hopefully straightforward question to ask concerning the following lovely version of Siegel's Lemma: Let $K$ be some algebraic number field, and let $O_K$ be it's ring of integers. Define ...
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31 views

Continuity of Function involving logarithm function

I want to prove a function $f(x) = g(x) * log x $ is continuous on interval $[0, 1]$, where value of $g (x)$ is $0$ at lower limit point $0$. Anybody can help me out here. Thanks in advance.
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83 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
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57 views

Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
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When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...
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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 ...
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183 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...
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1answer
81 views

Transcendence Degree of the Function Field of Meromorphic Functions over $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense ...
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2answers
513 views

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.
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107 views

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 ...
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1answer
133 views

Integers (strictly) between 0 and 1 form the basis of transcendental number theory?

In a MathOverflow comment on the question of "What is the most useful non-existing object of your field?", an answer is given A number which is less than 1 and greater than 1. Which elicited a ...
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1answer
36 views

Exceptional Set and Schanuel's conjecture

I was reading an article about transcendental funtions (Algebraic values of transcendental functions at algebraic points, by Huang, J., Marques, D., Mereb, M.). The authors gave an example that says: ...
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577 views

square-root of a transcendental number

I know that a square-root of an irrational number is also irrational. Is it also true that the square root of a transcendental number is transcendental?
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106 views

Are there classifications transcendental numbers that are similar to algebraic numbers for differential equations?

Considering that transcendental numbers are described as not a root of a non-zero polynomial equation with rational coefficients, are there classifications of transcendental numbers that are ...
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67 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
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Borel's result on transcendence measure

In "Sur la nature arithmétique du nombre e" (Comptes rendus de l'Académie des Sciences 128 (1899), 596-9) Borel presented his result on transcendence measure for e. This can be restated as follows: ...
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57 views

Relating sequence of integers with decimal representation

Let a sequence of integers be defined by a rule/formula,if this sequence is supposed to be a decimal representation of a number such as: $1)$ $a_{n+1}=a_n+1$ (given $a_0=0$) ...
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737 views

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 ...
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Irreducible polynomial in relatively algebraically closed extension

I am thinking about how to prove this fact: given a field extension $K\subseteq L$ such that $K$ is algebraically closed in $L$, and an irreducible polynomial $f\in K[X]$. Prove that $f$ is ...
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2answers
194 views

transcendental number theory - classification

On Wikipedia one can find that transcendental number theory, or transcendednce theory is a branch of number theory. That confuses me a little since I thought that number theory is concerned with ...
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146 views

irrationality measure

I was reading that you can associate a measure to any given number giving you "how irrational" the given number is. I was wondering is there any irrationality measure that would tell you that the ...
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2answers
60 views

Are elements of the range of a transcendental function themselves transcendental, excepting a “few” special cases?

Let $f(x)$ be a transcendental function with $x\in\mathbb{C}$. Then are the values $f(x)$ themselves transcendental, except perhaps for a "few" exceptions? For example, it is known that $f(x)=e^x$ is ...
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1answer
519 views

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...
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1answer
451 views

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?
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329 views

Understanding the Fundamental Theorem of Symmetric Polynomials within the context of proving $\pi$ transcendental

I am currently studying the proof of the transcendence of $\pi$. There are a bunch of proofs scattered across the web (here, here, and here, to list some); some derive from the Lindemann-Weierstrass ...
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1answer
72 views

Operations with cardinals in transcendence base proof

I want to prove the theorem that two transcendence bases for a transcendental field extension have the same cardinality. I've come with this situation : if $A$ and $B$ are two transcendence bases for ...
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248 views

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 ...
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1answer
239 views

Is it transcendental? Also normal?

The number we are considering is as follows: $0.a_1 a_2 a_3 \cdots $, where $a_{2n-1}=(n)_{(2)}, a_{2n}=(n)_{(3)}.$ So, the number is $$0.(1)(1)(10)(2)(11)(10)(100)(11)(101)(12)\cdots.$$ Is the ...
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1answer
449 views

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 $2$?
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1answer
88 views

Representing complex numbers with nested exponentiation of rationals

Define $L_0=Q$ $L_1=\lbrace x \in C; e^{x} \in L_0 \rbrace$ $L_{-1}=\lbrace x \in C; \ln{x} \in L_0 \rbrace$ $L_{n+1}=\lbrace x \in C; e^{x} \in L_n \rbrace$ $0$ is in $L_1$ and $L_0$. Do any ...
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1answer
85 views

Do the first differences of a b-normal number form a b-normal number? [Now with application!]

If it true that for every $b$-normal number (where the base $b,b^2,b^3,\ldots$ digits are asymptotically equiprobable) that the first differences of the base-$b$ digits, interpreted as a (signed) ...
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3answers
6k views

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, ...
3
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2answers
262 views

Is there a proof that there is no general method to solve transcendental equations?

Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems ...