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 ...
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1 vote
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Given a partition of a field, construct a partition of its extension

The motivation for my question is the following algebraic consequence of the Continuum Hypothesis ($2^{\aleph_0}=\aleph_1$) by Zoli: (T1) Assume the Continuum Hypothesis holds. Then $\mathbb{R}^{\...
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6 votes
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Finite set of numbers whose powers sum up to irrational number

It is well-known that $e/\sqrt{2}$ is irrational. Indeed, if it was rational, i.e. $p/q$ then $e^2/2 =p^2/q^2.$ Thus, $q^2e^2=2p^2,$ which would imply that $e$ is a root of $q^2x^2=2p^2.$ Now my ...
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15 votes
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303 views

Transcendence of sum of reciprocals of factorials

For $A \subseteq \mathbb{N}$, define $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$. It is easy to see that for every infinite $A$, $x_A$ is irrational. Question: Is there an infinite $A \subseteq \...
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19 votes
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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: ...
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2 votes
1 answer
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 $$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$ ...
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8 votes
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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 ...
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3 votes
0 answers
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 ...
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  • 1,201
0 votes
2 answers
258 views

Transcendence on $ \alpha+f(\alpha), \alpha f(\alpha) $ and $ \alpha/f(\alpha) $ where $ \alpha$ is transcendental

Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\...
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13 votes
2 answers
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Why is it easier to prove $e$ is transcendental than $\pi$?

Why is it easier to prove $e$ is transcendental than $\pi$? I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
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6 votes
1 answer
87 views

Transcendence of values of Fredholm series at algebraic arguments

Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
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3 votes
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Does logarithms conjecture imply Gelfond conjecture?

Assume that logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(a_1),\cdots,\log(a_n)$ are ...
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12 votes
1 answer
557 views

Testing whether $e^x+ax^2+bx+c$ has a zero

What is the simple test with exponential polynomials to determine whether $$f(x)=e^x+ax^2+bx+c$$ has a positive zero? This was prompted by the question about discriminants here. We have an ineffective ...
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30 votes
1 answer
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How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?

I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation \begin{equation}\label{eq} x^{x+1}=(x+1)^x \end{equation} Let us define that ...
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4 votes
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187 views

On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$

For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number. Question. Are the two numbers $$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
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1 vote
0 answers
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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 ...
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6 votes
1 answer
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 ...
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17 votes
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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 ...
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3 votes
0 answers
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 ...
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0 answers
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Transcendence à la Liouville

Let $\alpha\in\overline{\mathbb Q}^*$. How to prove that for a non ultimately constant sequence $(\varepsilon_n)_n$ with $\varepsilon_n\in\{0,1\}$, the number $\sum_{n\ge0}\frac{\varepsilon_n}{\alpha^{...
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8 votes
1 answer
336 views

Irrationality measure of powers

Let $\alpha$ be an irrational number. Denote by $\mu(\alpha)$ its irrationality measure. Can one say anything about $\mu(\alpha^n)$ for every $n\in\mathbb N$? Even more, one knows that $\mu(e)=2$. Can ...
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1 vote
1 answer
123 views

Hybrid numeration system on $[0,1]^2$

Let $X_0,X_1\in [0,1]$ and $b_1,b_2>0$ be integers. We are going to create a numeration system for vectors $(X_0,X_1)$, the base being the vector $(b_1,b_2)$, as follows. Recursively define $X_k=\{...
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0 votes
0 answers
98 views

Transcendence of Euler series

Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
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  • 3,079
1 vote
0 answers
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 ...
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2 votes
0 answers
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 $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
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  • 3,079
4 votes
1 answer
361 views

About $\pi$, $e$ and transcendence

This is mostly curiosity on my part. I assume experts would have some up-to-date info. Are $1$, $\pi$ and $e$ linearly independent over $\mathbb{Q}$? Does the set $\{ m\pi+ne;\;\;m,n\in\mathbb{Z}\...
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3 votes
0 answers
140 views

Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
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3 votes
0 answers
157 views

What is known about the irrationality of ratios and products of logarithms of integers?

Let $a,b,c,d$ be positive natural numbers such that $\{a,b\} \neq \{c,d\}$ and such that none are perfect powers. Is it true that $$\frac{\log a \log b}{\log c \log d} \notin \mathbb{Q} ?$$ The ...
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6 votes
1 answer
578 views

Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...
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12 votes
1 answer
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 ...
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1 vote
0 answers
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 $\log{n}$? ...
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-6 votes
1 answer
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 ...
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6 votes
1 answer
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 ...
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0 votes
1 answer
152 views

Any ideas for the following limit of partial sums of binomial coefficients?

Let $a$ be an odd integer $≥3$. It appears that: $$\lim_{n\rightarrow\infty}\frac{1}{2^{n}}\sum_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases} 1 & \textrm{if }a=3\...
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6 votes
0 answers
89 views

Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator

This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
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3 votes
1 answer
197 views

Power series equation with solution $1/e$ [closed]

As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root. Are there classical equations of the form $$\sum_{i=0}^{\infty} a_ix^i =1$$ that have $e$ ...
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3 votes
0 answers
65 views

Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$

Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and $$ P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9}, $$ a Fourier series of quasi-modular form ...
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  • 511
3 votes
1 answer
190 views

Linear independence of approximate square roots

From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
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  • 263
1 vote
0 answers
556 views

Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]

Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José). My question is, can we ...
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2 votes
0 answers
132 views

Compare my software's representation of exponential numbers and 0?

Suppose I have a real number $$ x=\sum_{i=1}^n a_i e^{\lambda_i} $$ where $a_i,\lambda_i$s are complex algebraic numbers. Is there an algorithm to determine whether it is greater than 0 or less than ...
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  • 1,265
4 votes
1 answer
225 views

Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
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1 vote
0 answers
122 views

Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true

The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$: Theorem (Hermite-Lindemann-Weierstrass): Let $\...
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1 vote
1 answer
136 views

Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?

I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here. The Prouhet-Thue-Morse constant, defined as $$ \tau =...
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0 votes
0 answers
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 ...
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7 votes
1 answer
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 ...
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  • 415
1 vote
1 answer
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 ...
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9 votes
0 answers
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 ...
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  • 191
12 votes
1 answer
846 views

Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...
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  • 1,216
1 vote
0 answers
95 views

unknown sequences of rational numbers with sum of a transcendental number [closed]

Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form : $e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{...
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  • 269
5 votes
1 answer
430 views

Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
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