# Questions tagged [transcendental-number-theory]

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### Markov constant of $\pi$

Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$ According to this document, if ...
1 vote
77 views

531 views

### Hensel's proof that $e$ is transcendental

When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: ...
104 views

### Algebraicity of a generating function and binomial numbers

It is an obvious fact that the sum $\sum_{n\geq 0} \binom{2n}{n} x^n$ defines an algebraic function. I am interested in the variation of this sum, namely $$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$ ...
148 views

### Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
130 views

### Transcendentality of Coleman integral

I wonder if there's any work that considers the algebracity/transcendentality of Coleman integral (over $\mathbb{Q}_{p}$). The reason I think about this is because, for hyperelliptic curves, there are ...
258 views

1 vote
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### Is the “amalgamation” of an enumerated infinite collection of absolutely normal real numbers always absolutely normal?

Let $S$ denote an enumerated infinite collection of absolutely normal (i.e. normal in all integer bases greater than or equal to $2$) real numbers (with no repetitions) such that any natural number ...
528 views

### Do we have an algorithm for comparing $e^e$ with rationals?

Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence? In a non-constructive sense, there obviously is an algorithm. If $e^e$ is some rational $q_0$, then we ...
584 views

### Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
306 views

### Galois theory of periods of algebraic varieties PhD project

I finished my MMath at Exeter in July of this year and I would like to undertake a PhD at the interface of algebraic geometry, number theory and Galois theory. More specifically, I recently came ...
83 views

98 views

### Transcendence of Euler series

Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
1 vote
61 views

### Transcendence detection from algebraic constructions

I have a simple image or intuition in my mind that I can't seem to shake off, so I thought I'd seek help here. Suppose we don't know if $\alpha \in \mathbb{C} - \mathbb{Q}$ is algebraic or ...
74 views

### Transcendence and Mahler's method

Let $\alpha,\beta\in\overline{\mathbb Q}$ be such that $0<|a|<1$ and $|\beta|=1$. One assumes that $\beta$ is not a root of unity. Is $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
361 views

480 views

### Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
1 vote
91 views

### Efficiently computing the digits of irrational number

Is there irrational real number $C$, defined by algebraic numbers and elementary functions such that the $n$-th digit in base $b$ in the fractional part is computable in time polynomial in $\log{n}$? ...
283 views

### Numerical evidence that $\pi$ is not normal in base two [closed]

Confusion is possible, but we got numerical evidence against popular belief about the normality of $\pi$ in base two. According to wikipedia a real number is said to be simply normal in an integer ...
324 views

### Algorithm for comparing $x + y \cdot \log(z)$ for $x,y,z$ rational?

I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures ...
152 views

209 views

### Transcendental functions generating almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
593 views

### Digits in an algebraic irrational number

I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture). I know that by using Ridout theorem or Schmidt subspace theorem ...
1 vote
524 views

### Closest area of research to Transcendental Number Theory or/and Geometry of Numbers?

I am highly interested in doing research in either of 1- Transcendental Number Theory and Algebraic Independence; 2- Diophantine Approximation and Geometry of Numbers. There is no person working ... 250 views

### Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...