Questions tagged [transcendental-number-theory]
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133
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3
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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 ...
- 205
1
vote
0
answers
77
views
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}^{\...
- 131
6
votes
1
answer
424
views
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 ...
- 315
15
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0
answers
303
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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 \...
- 151
19
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1
answer
531
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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: ...
- 9,500
2
votes
1
answer
104
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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
$$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$
...
- 1,054
8
votes
0
answers
148
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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 ...
- 449
3
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0
answers
130
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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 ...
- 1,201
0
votes
2
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258
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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 $$\alpha+f(\alpha),\...
- 343
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2
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1k
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Why is it easier to prove $e$ is transcendental than $\pi$?
Why is it easier to prove $e$ is transcendental than $\pi$?
I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
- 307
6
votes
1
answer
87
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Transcendence of values of Fredholm series at algebraic arguments
Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
- 3,079
3
votes
0
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80
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Does logarithms conjecture imply Gelfond conjecture?
Assume that logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(a_1),\cdots,\log(a_n)$ are ...
- 3,079
12
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1
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557
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Testing whether $e^x+ax^2+bx+c$ has a zero
What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...
- 16.7k
30
votes
1
answer
1k
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How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...
- 1,235
4
votes
0
answers
187
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On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$
For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number.
Question. Are the two numbers
$$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
- 13k
1
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0
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48
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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 ...
- 695
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528
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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 ...
- 16.7k
17
votes
0
answers
584
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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 ...
- 64.8k
3
votes
0
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306
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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 ...
- 31
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83
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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^{...
- 3,079
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votes
1
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336
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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 ...
- 3,079
1
vote
1
answer
123
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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=\{...
- 2,415
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0
answers
98
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Transcendence of Euler series
Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
- 3,079
1
vote
0
answers
61
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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 ...
- 19
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0
answers
74
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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 $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
- 3,079
4
votes
1
answer
361
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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}\...
- 32.3k
3
votes
0
answers
140
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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 $W(1)=\Omega$ ...
- 55
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0
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157
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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 ...
- 10.9k
6
votes
1
answer
578
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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)=...
- 1,728
12
votes
1
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480
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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 ...
- 13.2k
1
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0
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91
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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 $\log{n}$?
...
- 23.4k
-6
votes
1
answer
283
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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 ...
- 23.4k
6
votes
1
answer
324
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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 ...
- 12.5k
0
votes
1
answer
152
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Any ideas for the following limit of partial sums of binomial coefficients?
Let $a$ be an odd integer $≥3$. It appears that: $$\lim_{n\rightarrow\infty}\frac{1}{2^{n}}\sum_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases}
1 & \textrm{if }a=3\...
- 1,216
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votes
0
answers
89
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Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator
This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
- 55
3
votes
1
answer
197
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Power series equation with solution $1/e$ [closed]
As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root.
Are there classical equations of the form
$$\sum_{i=0}^{\infty} a_ix^i =1$$
that have $e$ ...
- 53
3
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0
answers
65
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Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$
Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and
$$
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},
$$
a Fourier series of quasi-modular form ...
- 511
3
votes
1
answer
190
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Linear independence of approximate square roots
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
- 263
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556
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Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]
Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...
user141903
2
votes
0
answers
132
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Compare my software's representation of exponential numbers and 0?
Suppose I have a real number
$$
x=\sum_{i=1}^n a_i e^{\lambda_i}
$$
where $a_i,\lambda_i$s are complex algebraic numbers.
Is there an algorithm to determine whether it is greater than 0 or less than ...
- 1,265
4
votes
1
answer
225
views
Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
- 1,728
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0
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122
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Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true
The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$:
Theorem (Hermite-Lindemann-Weierstrass): Let $\...
- 21.4k
1
vote
1
answer
136
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Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?
I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.
The Prouhet-Thue-Morse constant, defined as
$$
\tau =...
- 1,841
0
votes
0
answers
209
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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 ...
7
votes
1
answer
593
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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 ...
- 415
1
vote
1
answer
524
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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 ...
user135767
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 ...
- 191
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votes
1
answer
846
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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^{-\...
- 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}{...
- 269
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votes
1
answer
430
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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 ...
- 39.7k