The transcend.-number-theory tag has no usage guidance.

**38**

votes

**2**answers

1k views

### 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.
...

**32**

votes

**2**answers

936 views

### Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, i thought it was easy to show that there are not integer solutions to this equation, but then i realized that the ...

**29**

votes

**3**answers

885 views

### Unexpected applications of transcendental number theory?

In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of ...

**24**

votes

**1**answer

2k views

### More on “Transalgebraic Theories” (a 19th century yoga)?

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful ...

**22**

votes

**1**answer

1k views

### Is this seemingly novel irrational constant also transcendental?

I recently discovered a constant that is constructed as follows:
$\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$
Furthermore I can prove that it is an irrational number ...

**22**

votes

**0**answers

425 views

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...

**21**

votes

**2**answers

5k views

### 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?

**18**

votes

**1**answer

638 views

### Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...

**15**

votes

**2**answers

522 views

### 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| ...

**14**

votes

**1**answer

2k views

### “The Galois group of $\pi$ is $\mathbb{Z}$.”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what ...

**14**

votes

**0**answers

361 views

### Integer-valued power towers

$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$
Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$.
Are there any positive ...

**12**

votes

**5**answers

1k views

### 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 ...

**11**

votes

**2**answers

522 views

### 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 ...

**11**

votes

**1**answer

416 views

### Transcendence of products of certain real algebraic numbers

Let
\begin{equation}
z := \prod_p p^{1/p^2},
\end{equation}
where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...

**11**

votes

**0**answers

521 views

### 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 + ...

**10**

votes

**1**answer

827 views

### 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, ...

**10**

votes

**1**answer

874 views

### 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) ...

**10**

votes

**0**answers

501 views

### Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.
A famous result of Polya says if $f$ is an entire function of ...

**9**

votes

**1**answer

583 views

### Is $e^p\in\mathbb{Q}_p$ known to be transcendental?

$\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}$ doesn't converge in $\mathbb{Q}_p$, however, $e^p:=\sum\limits_{n=0}^{\infty}\dfrac{p^n}{n!}$ does converge for $p\neq 2$. So my question is,
Are ...

**8**

votes

**1**answer

426 views

### 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. ...

**7**

votes

**5**answers

2k views

### Advice on choosing an area of specialization

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 ...

**7**

votes

**5**answers

792 views

### Small values of a polynomial evaluated at roots of unity

The MO answer http://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant ...

**5**

votes

**2**answers

626 views

### 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?

**5**

votes

**3**answers

3k views

### Transcendence of PI

Can anyone suggest me an ingenious proof of the transcendence of $\pi$. I have seen Lindemann's proof but it appears intricate.

**5**

votes

**2**answers

580 views

### A transcendence question involving the exponential function

Let $(z_n)$ be a sequence of complex numbers satisfying $|z_n|\to +\infty$ and such that $\{e^{z_n}\mid n \in \mathbb{N}\}$ is infinite.
Is it always true that $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ ...

**5**

votes

**2**answers

687 views

### 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 ...

**5**

votes

**1**answer

389 views

### Can Liouville's number be expressed as a physical ratio in the sense that $\pi$ is?

Quadratic irrational numbers are perhaps the most basic examples of irrational numbers that arise as basic physical ratios: think of $\sqrt{2}$ as the distance between the corners of a square to the ...

**5**

votes

**1**answer

567 views

### Are these numbers irrational and/or transcendental?

I am curious to know if the following number is irrational or transcendental:
$$\displaystyle A = \sum_p 2^{-p},$$
where the sum is over all positive primes. A similar question can be asked for any ...

**5**

votes

**1**answer

609 views

### Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...

**5**

votes

**5**answers

1k views

### Background Reading for Proving Irrationality of Real Numbers

I'm almost finishing my PhD in applied mathematics, but I'm planning soon (after doing post-doc) to start seriously doing research on problems about proving irrationality of
real numbers. Whenever I ...

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votes

**1**answer

202 views

### transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...

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**0**answers

100 views

### On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...

**4**

votes

**1**answer

798 views

### Strongest known version of Baker's theorem

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question:
What is the the strongest known lower bound for ...

**4**

votes

**1**answer

206 views

### Transcendence of a ratio of p-adic logarithms

Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that ...

**4**

votes

**2**answers

486 views

### Is there a reasonable definition of the height of a transcendental number

For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its ...

**4**

votes

**1**answer

177 views

### 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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votes

**0**answers

306 views

### A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...

**3**

votes

**1**answer

245 views

### 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 ...

**3**

votes

**1**answer

270 views

### Identity between Euler gamma and pi

Let $\gamma$ be Euler constant and $W$ Lambert W function.
One can show:
$$-2/3\,{\frac {\gamma+\ln \left( \pi \right) }{W \left(
-1/3\,{\frac { \left( \gamma+\ln \left( \pi \right) \right) {
...

**3**

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**0**answers

129 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

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votes

**0**answers

182 views

### 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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**0**answers

72 views

### Transcendence of a $q$-series

Let $q\ge 2$ be an integer. Fourier's proof of irrationality of $e$ adapts to prove the irrationality of
$$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$
Is this number knwon to be ...

**2**

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**0**answers

115 views

### 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 ...

**2**

votes

**0**answers

254 views

### schanuel's conjecture and real root of $x+e^x=0$

Schanuel's conjecture states:
-If $\alpha_1,\alpha_2,...,\alpha_n$ are complex numbers linearly independent over $\mathbb{Q}$ then the trascendence degree of the field ...

**1**

vote

**1**answer

274 views

### Rational points of non-rational curves

An algebraic curve (in this question) is the zero set $C = f^{-1}(X\ Y)$ of any polynomial $f\in\mathbb R[X\ Y]$; we say then that $f$ represents $C$. ...

**1**

vote

**1**answer

491 views

### How can we understand Baker's theorem about transcendence ?

We know that for algebraic $e^{it\pi}$, $t$ can not be algebraic irrational by
Baker's theorem, but his proof is analytic; is there some algebraic understanding for such fact? If $t$ is ...

**1**

vote

**1**answer

398 views

### Minimal Polynomials for Algebraic Dependence?

Hello all!
I recently had a question concerning algebraic dependence that has thus far gone unanswered from my professors and texts, that I hope I can phrase properly here. When answering, please ...

**0**

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**1**answer

435 views

### Does this isomorphism between Galois groups hold for transcendental extensions?

Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have:
$$\text{Gal}(L/K)\cong ...

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134 views

### Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$
$$P_2(x,y_1,\dots,y_n)=0,$$
$$\vdots$$
$$P_k(x,y_1,\dots,y_n)=0$$
is a system of equations with coefficients over $\mathbb{Z}$, and ...

**0**

votes

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176 views

### 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 ...