# Tagged Questions

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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### Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five ...
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### Deducing Lindemann-Weierstrass from Baker's theorem

I'm aware that Baker's theorem with $n=1$ (for one algebraic number only) follows from that of Lindemann-Weierstrass. It is also often mentioned that Baker's result is a generalization of Lindemann-...
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### 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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### 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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### 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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### 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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### 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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### 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 Gelfondâ€“...
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### 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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### 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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### 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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### 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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### 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 arithmĂ©tique du nombre e" (Comptes rendus de l'AcadĂ©mie 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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### 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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### 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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### 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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### 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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### 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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### Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfondâ€“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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### 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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### 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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### 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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### 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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### 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) base-...
### 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, ...