All Questions
Tagged with irrational-numbers nt.number-theory
71 questions
12
votes
1
answer
565
views
On Bailey–Borwein–Plouffe formula for irrational numbers
A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...
1
vote
0
answers
176
views
binomial coefficients and irrationals
The following, probably either currently impossible to deal with, or
having a negative solution, arose from an ergodic theory question,
presumably itself currently intractible. I am not a number ...
- 4,679
14
votes
1
answer
1k
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 ...
- 2,249
6
votes
0
answers
465
views
Conway's box function iterated to produce a hierarchy of nested sets of real numbers
Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
2
votes
0
answers
148
views
Bitwise operation of two square roots
Let $\sqrt 2 = 1.a_1a_2\dots _2$, and $\sqrt 3 = 1.b_1b_2\dots _2$. What can one say about the number $n = 0.c_1c_2\dots$ where $c_i = 1$ if $a_i = b_i$ and $0$ otherwise? There is no reason to ...
- 1,890
33
votes
2
answers
1k
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 no non-trivial integer solutions to this equation, but then I ...
- 515
2
votes
1
answer
88
views
On finding the region $R$ for which the multi-variable sequence converges [closed]
Find the region $(x,y) \in R$ for which the following sequence converges
$$\lim_{n \to \infty} \; \;\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| = 0$$
I am currently doing number theory ...
- 1,549
25
votes
1
answer
2k
views
Question on the irrationality of $e$
I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...
- 1,549
19
votes
1
answer
939
views
Steinhaus's Easter Egg Problem
The following is the text of Steinhaus's so-called Easter egg problem. According to this article of Roman Duda, this was recorded in the New Scottish Book around Easter 1955 and "Steinhaus offered an ...
- 13k
25
votes
2
answers
4k
views
A "better" rational approximation of pi?
$355/113$ is a good fractional approximation of $\pi$, because we use six digits to produce seven correct digits of $\pi$.
$$\frac{355}{113} = 3.1415929\ldots$$
Let $R$ be the ratio of the number of ...
- 251
7
votes
1
answer
823
views
Irrationality of Dedekind zeta values
For Riemann's zeta function, one knows that:
$\zeta(2n)$ is irrational (because a rational multiple of $\pi^{2n}$ is)
$\zeta(3)$ is irrational (proved by Apéry)
and a few other results like "there ...
- 71
10
votes
2
answers
4k
views
Lebesgue measure of a set of irrational numbers
Let $I_{\lambda},$ $\lambda>0$ be a subset of all irrational numbers $\rho=[a_{1},a_{2},...,a_{n},...]\in(0,1)$ such that $a_{n}\leq \text{const}\cdot n^{\lambda}.$
Here, $[a_{1},a_{2},...,a_{n},.....
- 103
5
votes
1
answer
2k
views
Lebesgue measure of some set of irrational numbers
Let $(i_{n})$ be a strictly increasing sequence of natural numbers,
$(v_{n})$ be an unbounded sequences of natural numbers
and $M\geq 2$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all ...
- 197
10
votes
1
answer
3k
views
Measure of a set of irrational numbers
Let $A$ be a set of all irrational numbers $\rho \in (0, 1)$ represented as a continued fraction $\rho=[a_{1}, a_{2},...,a_{n},...],$ such that $a_{n}\leq \text{const}\cdot n^{\epsilon}$ for some $\...
- 197
6
votes
0
answers
306
views
Irrationality of the sum of the reciprocal of perfect powers
A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that $$\sum_{p\in\...
- 10.9k
-8
votes
1
answer
959
views
If $a$ is irrational, must $a^a$ be irrational? [closed]
It is known that $\sqrt{2}^{\sqrt{2}}$ is irrational. Is it true that for any irrational number $a$, $a^a$ must be irrational?
- 1,160
1
vote
1
answer
163
views
A question on subgroup-restricted irrationality measures
The irrationality measure μ(x) of a positive irrational number x is defined to be the supremum of the exponents e such that |x - p/q| < 1/q^e has an infinite number of solutions p/q. By the ...
- 679
22
votes
1
answer
11k
views
Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem
Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gelfond–Schneider theorem.
We know that ${\sqrt2}^{\sqrt2}$ is a transcendental number by the Gel'fond-Schneider's theorem. I'...
- 4,757
16
votes
5
answers
9k
views
Elementary proof of the equidistribution theorem
I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...
- 169
49
votes
2
answers
19k
views
Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$
What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
- 6,819
35
votes
9
answers
21k
views
Direct proof of irrationality?
There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form:
Suppose $a/b=\sqrt{2}$ ...
- 395