Questions tagged [transcendental-number-theory]
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Proof that pi is transcendental that doesn't use the infinitude of primes
I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...
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Work on independence of pi and e
It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.
What are some of the important results leading toward proving this?
What are the most promising ...
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If $2^x $and $3^x$ are integers, must $x$ be as well?
I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
If $n^x$ is an integer for ...
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Why is it hard to prove that the Euler Mascheroni constant is irrational?
Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?
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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 ...
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Advances and difficulties in effective version of Thue-Roth-Siegel Theorem
A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...
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Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?
Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?
Is there a survey with up-to-date transcendence results?
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Advice on choosing an area of specialization [closed]
I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...
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On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$
Pick integers $a, b \ge 2$ and let $\xi_{a,b}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n}$. It is known that $\xi_{2,2}$ is transcendental: I learned a proof of this from notes by M. ...
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Conjecture on irrational algebraic numbers
Conjecture:
For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
Questions:
Has this conjecture been ...
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To prove irrationality, why integrate?
I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of ...
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Numbers that are generic w.r.t. exponentiation
This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways.
...
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Proving the irrationality of $\pi e$ and $\pi / e$
Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
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Numbers with known finite irrationality measure greater than 2
For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that
$$ \left| \alpha-\frac{p}{q}\right|...
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Transcendence of $\Gamma(1/3), \Gamma(1/4)$
This is a re-post from MSE as I did not get even a single comment there.
Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
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For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?
It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
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Transcendental numbers: yet another classification
Let $\mathbb{A^+}$ be the set of non-negative algebraic numbers. Consider the set of "polynomials" : $$\mathbb{P} = \lbrace a_0 + a_1x^{r_1} + a_2x^{r_2} + a_3x^{r_3} +\cdots + a_nx^{r_n}| a_0, a_i, ...
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Connection between Infinite continued fractions, elliptic integrals and AGM
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
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Testing whether $e^x+ax^2+bx+c$ has a zero
What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...
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Degree of Transcendentality and Feynman Diagrams
Physicists computing multiloop Feynman diagrams have introduced various
techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines
1) ...
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Constructing an infinite chain of subsets of 'hyper' algebraic numbers?
This question is cross posted from MSE.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...
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Algebraic independence of $a^a$ and $b^b$ for algebraic irrationals $a,b$
Let $a,b$ be algebraic irrationals.
Are there conjectures or unconditional results about the algebraic
independence of $a^a$ and $b^b$?
Probably Schanuel's conjecture is related,
maybe only $\log{a},...
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Markov constant of $\pi$
Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$
According to this document, if ...
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1
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Hermite Lindemann and transcendental reals
This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals.
Anyway, let $E$ be the "constructible numbers," ...
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Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$
Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers.
Question 1: Is it true that $\mathbf{C}$ is ...
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Transcendency of certain integrals
Let $p$ be an even monic polynomial with rational coefficients of degree at least $4$.
Can the integral
$$\int_{-\infty}^\infty e^{-p(x)}dx$$
be an algebraic number? Is anything known about ...
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1
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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=\{...
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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 ...
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algebraic independence of exponential functions
Let $a_1,\ldots,a_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)?
I ...
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On the irrationality measure of generalized Stoneham numbers
Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and $\gcd(a,...