# Tagged Questions

**2**

votes

**0**answers

216 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 ...

**17**

votes

**1**answer

522 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 ...

**2**

votes

**1**answer

212 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) {
...

**0**

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**0**answers

142 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 ...

**7**

votes

**1**answer

346 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. ...

**1**

vote

**1**answer

219 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 $C$. ...

**13**

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**0**answers

325 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 ...

**37**

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.
...

**10**

votes

**2**answers

387 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| ...

**11**

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**2**answers

2k 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?

**1**

vote

**1**answer

376 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 ...

**4**

votes

**1**answer

460 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 ...

**6**

votes

**1**answer

424 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 ...

**20**

votes

**1**answer

988 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 ...

**5**

votes

**0**answers

740 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 ...

**4**

votes

**2**answers

459 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**

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**0**answers

272 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 ...

**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 ...

**3**

votes

**2**answers

504 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?

**4**

votes

**1**answer

740 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 ...