Intereting Posts

Exchange order of “almost all” quantifiers
Expected number of runs
Finding Required Permutation
Exercise 3.39 of Fulton & Harris
The number of positive integers whose digits are all $1$, $3$, or $4$, and add up to $2k$, is a perfect square
The comprehension axioms follows from the replacement schema.
Action of a group on itself by conjugation is faithful $\iff$ trivial center
Find $n$ for which $\frac{n(n+1)}{2}$ is perfect square
Can Path Connectedness be Defined without Using the Unit Interval?
Axiom of extensionality in ZF – pointless?
Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
How to show that ${2},\sqrt{3}):\mathbb{Q}]=9$?
$\{m \alpha, m \in \mathbb Z\}$is dense in $$ for $\alpha$ irrational
How do I evaluate this limit: $\lim_{n\to+\infty}\sum_{k=1}^{n} \frac{1}{k(k+1)\cdots(k+m)}$?
Sums of two perfect squares

Let $u$ be the solution of the equation $$e^x \log(x)=1$$

Is $u$ rational, irrational algebraic or transcendental?

$u$ seems to be transcendental, but I cannot prove it.

- Is it known if $\pi + e$ is transcendental over the rational numbers?
- if $x\ne 0$, is at least one of $\{x, \cos\;x\}$ transcendental over $\mathbb{Q}$?
- Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?
- Sum and Product of two transcendental numbers cannot be simultaneously algebraic
- Sum and product of two transcendental numbers can't be both algebraic
- Does there exist $\alpha \in \mathbb{R}$ and a field $F \subset \mathbb{R} $ such that $F(\alpha)=\mathbb{R}$?

Perhaps, someone has an idea.

- “The Galois group of $\pi$ is $\mathbb{Z}$”
- What is the probability that $250$ random digits contain $7777$ , $8888$ and $9999$?
- How was the difference of the Fransén–Robinson constant and Euler's number found?
- if $x\ne 0$, is at least one of $\{x, \cos\;x\}$ transcendental over $\mathbb{Q}$?
- Proof that at most one of $e\pi$ and $e+\pi$ can be rational
- Deciding whether $2^{\sqrt2}$ is irrational/transcendental
- For integer $k > 1$, is $\sum_{i=0}^{\infty} 1/k^{2^i}$ transcendental or algebraic, or unknown?
- Proof that $e^x$ is a transcendental function of $x$?
- Producing infinite family of transcendental numbers
- Is it known if $\pi + e$ is transcendental over the rational numbers?

- A strongly pseudomonotone map that is not strongly monotone
- How to determine the difference Onto vs One-to-one?
- Can a basis for $\mathbb{R}$ be Borel?
- Solve the sequence : $u_n = 1-(\frac{u_1}{n} + \frac{u_2}{n-1} + \ldots + \frac{u_{n-1}}{2})$
- Eigenvalues and eigenvectors in physics?
- 100 prisoners and a lightbulb
- What does a Godel sentence actually look like?
- Equivalent definitions of the Jacobson Radical
- Why can't we define more elementary functions?
- Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.
- An application of partitions of unity: integrating over open sets.
- Boundedness of operator on Hilbert space
- Prove$\overline{(A \cap B \cap C)} = \overline{A} \cup \overline{B} \cup \overline{C}$ By Subsets
- Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?
- Every Group is a Fundamental Group