12
votes
0answers
153 views

Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
0
votes
2answers
65 views

Lebesgue measure of transcendental numbers in $[0,1]$.

What is the Lebesgue measure of the transcendental numbers in the $[0,1]$ interval? I was not able to find any information on this. (Does this question even make sense given what we currently know ...
14
votes
3answers
531 views

Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
3
votes
0answers
20 views

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$? If $a$ is transcendental over ${\mathbb Q}[\alpha]$ then the ...
3
votes
1answer
95 views

The set $\{\frac{3^m}{\alpha^n}:\;m,n\in\mathbb Z\}$ is dense in $\mathbb R_+$

This seems that the set $$\left\{\frac{3^m}{\alpha^n}:\;m,n\in\mathbb Z\right\}$$ is dense in $\mathbb R_+$ (the set of positive real numbers), but I can not find the proof. How to prove this? Edit ...
4
votes
0answers
438 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...