All Questions
Tagged with irrational-numbers nt.number-theory
71 questions
67
votes
2
answers
6k
views
To prove irrationality, why integrate?
I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of ...
- 82.7k
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 $e^{\pi^2}$?
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40
votes
5
answers
3k
views
The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic
Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$:
$0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$.
Prove that the sequence $(a_{n})$ is periodic.
This ...
- 3,153
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
35
votes
1
answer
3k
views
Proving the irrationality of $\pi e$ and $\pi / e$
Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
- 1,549
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
30
votes
1
answer
1k
views
How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...
- 1,343
29
votes
5
answers
3k
views
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
- 1,962
26
votes
0
answers
841
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 ...
- 3,028
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
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
22
votes
2
answers
2k
views
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
Is it known whether $$\int_0^1 e^{-x^2} \, dx$$ is irrational? It is well-known that $\int_0^\infty e^{-x^2} \, dx=\frac{\sqrt{\pi}}{2}$ which is irrational, but what about the prior integral? Also, I ...
- 323
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
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
17
votes
3
answers
718
views
Fractional part power
Does a irrational number $x > 1$ exist such that $\{x^n \} \le \frac{1}{2}$ for
all positive integers $n$ ?
$x=1+ \sqrt 2$ holds for $n$ odd, but not in even
- 683
17
votes
0
answers
743
views
Picture of Lambert's proof that $\pi$ is irrational?
With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
- 82.7k
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
16
votes
1
answer
2k
views
Extending Apéry's proof to Catalan's constant?
I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction?
$$\begin{equation*} \zeta (3)=...
- 533
14
votes
1
answer
1k
views
Transcendence of $\Gamma(1/3), \Gamma(1/4)$
This is a re-post from MSE as I did not get even a single comment there.
Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
- 2,249
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
12
votes
1
answer
1k
views
Is $e^{{e^{\ \dots\ }}^n}$ ever an integer?
Let $n$ be a positive integer. It is clear that $e^n$ is not integer because $e$ is transcendental (not algebraic).
Now for each positive integer $k$ let $F^k(n)$ denote the $k$-fold composition of $F(...
- 3,317
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 ...
11
votes
4
answers
1k
views
Compilation of strategies to show that some constant is irrational
I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
- 521
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
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
8
votes
2
answers
340
views
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
- 3,098
8
votes
2
answers
627
views
Irrationality measure of arctan(1/3)
I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: ...
- 251
8
votes
3
answers
707
views
Irrationality of generalized continued fractions
An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z\setminus\left\{0\right\})$$
is irrational. Now for a generalized continued fraction:
$$\...
- 201
8
votes
2
answers
387
views
Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain.
Is it possible to estimate ...
user215601
8
votes
1
answer
765
views
An alternative to continued fraction and applications
This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...
7
votes
1
answer
480
views
An asymptotic formula in Apéry's proof of the irrationality of $\zeta(3)$
Let $a_n$ be the Apéry sequence
$$
a_n = \sum_{0\leq k\leq n}\binom{n}{k}^2\binom{n+k}{k}^2.
$$
Reading the 1978 paper Démonstration de l’irrationalité de $\zeta(3)$ (d’après R. Apery) of Cohen, at ...
- 271
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
7
votes
1
answer
400
views
Upper bounds on the irrationality measure of the arctan of an algebraic number
Let $x$ be an algebraic number. Must $\arctan(x)/\pi$ have finite irrationality measure? Are there any useful upper bounds?
- 547
7
votes
0
answers
621
views
Irrationality of the values of the prime zeta function
Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.
Since Apéry we know that $\zeta(3)$, ...
- 1,962
6
votes
1
answer
649
views
Algebraic and rational parts of a real number
Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers ...
- 3,098
6
votes
0
answers
283
views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
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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 ...
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 $$\sum_{p\in\...
- 10.9k
5
votes
1
answer
2k
views
Lebesgue measure of some set of irrational numbers
Let $(i_{n})$ be a strictly increasing sequence of natural numbers,
$(v_{n})$ be an unbounded sequences of natural numbers
and $M\geq 2$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all ...
- 197
5
votes
0
answers
109
views
Approximation of an irrational point from a given direction
Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
- 305
4
votes
1
answer
3k
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Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?
I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as ...
4
votes
0
answers
447
views
The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd
This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here.
We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
4
votes
0
answers
239
views
A connection between basic hypergeometric series and number theory
I am studying functions given by the power series:
$$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$
The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
- 2,188
3
votes
2
answers
452
views
Example of irrational number with a pattern in digits [closed]
Suppose I created the following random number generator.
A trusted person choose a irrational number. That can easily defined and computed by a computer. Like square root of a prime.
Every time the ...
- 179
3
votes
1
answer
315
views
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
- 3,098
3
votes
2
answers
1k
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Irrational number with known probability distribution on digits
Is there any irrational number that is known the probability distribution of digits?
Something like 0 appears 10% of time, 1 appears 10% of time, etc.
Probably irrational numbers that are defined ...
- 179
3
votes
1
answer
513
views
Regarding the digit expansion of $\sqrt 7$
Let $\sqrt 7=\sum_{i=0}^\infty a_i 7^{-i}, 0\le a_i \le 6$ be the expansion of $\sqrt 7$ in base $7$.
I am curious about the following question: Is there a $K\in \mathbb{N}$ such that for any $n\ge ...
- 31
3
votes
1
answer
208
views
$\psi(2,1/6),\psi(4,1/6)$ in terms of zeta and pi only and another closed form for zeta
Let $\psi(n,x)$ denote the polygamma function.
In this answer Lucia gave linear relations for $\psi(m,1/3),\psi(m,1/6),\zeta(m+1)$.
The computer managed to find closed form for $\psi(2,1/6)$ and $\...
- 25.4k
3
votes
1
answer
82
views
Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation
Happy New Year, MO community!
We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem.
PROBLEM
...
- 125
3
votes
1
answer
312
views
Looking for a proof that $\pi$ is irrational using a series representation for it
This have been asked on MSE but got no answers.
I'm searching for a proof that $\pi$ is irrational using a series representation for $\pi$, but can't find it.
However, on this wikipedia page show'...
- 521