48
votes
Accepted
How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
The number $x$ is transcendental, and your Gelfond-Schneider argument almost works.
Suppose to the contrary that $x$ is algebraic. Then $x+1$ and $x/(x+1)$ are also algebraic, and so the Gelfond-...
- 20.4k
48
votes
To prove irrationality, why integrate?
Hermite's approximations to values of $e^x$ are based on good rational function approximations to $e^x$, which nowadays go under the name of Padé approximations (a name that came much later: Padé was ...
- 50.6k
34
votes
Accepted
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
Following @te4's comment, we can look at the function $$f(x) = \frac{\sqrt{\pi}}{2\sqrt x} \operatorname{erf}(\sqrt{x}) = \sum_{n=0}^\infty \frac{(-1)^n x^n}{(2n+1) n!}.$$ Note that it's an E-function,...
- 3,319
26
votes
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
Regarding your second question, Apéry's amazing formula
$$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$
has inspired the search for analogous formulas for other zeta ...
- 82.6k
26
votes
Accepted
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
The proof of irrationality of $\displaystyle\sum_{n=0}^{+\infty}\frac1{F_n}$ (where $F_n$ is the $n$-th Fibonacci number) by RIchard André-Jeannin is an adaptation of the original Apery's proof of the ...
- 3,996
23
votes
Accepted
To prove irrationality, why integrate?
Here's an exposition of Niven's proof that makes the connection to orthogonal polynomials explicit. We start with an observation, easily proven by induction, that if $P\in \mathbb{Z}[x]$, then $\int_0^...
- 8,992
21
votes
Is $e^{{e^{\ \dots\ }}^n}$ ever an integer?
The impossibility of this would follow from Schanuel's conjecture but I would be surprised if it was known unconditionally. Let $q$ be rational and let $e_k = \exp^k(q)$, so that $e_0 = q$. We will ...
- 118k
21
votes
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
$\newcommand{\Q}{\mathbb Q}\newcommand{\erf}{\operatorname{erf}}$(This answer had been posted before I saw Command Master's answer. I am leaving it here, since it contains more and/or different ...
- 128k
20
votes
Accepted
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
The orbit of $1/3$ is infinite. You can show this via the $3$-adic valuation $\nu_3$.
Let us show by induction that $\nu_3(x_n) = -2^{n}$:
We have $\nu_3(x_0) = \nu_3(1/3) = -1 = -2^0$.
Now $\nu_3(x_{...
- 216
18
votes
Accepted
Why is the Euler-Mascheroni constant not a Liouville number?
At the present time, we do not even know how to prove that the Euler-Mascheroni constant $\gamma=\lim_{n\to\infty} \sum_{k=1}^n\frac{1}{k} - \log n$ is irrational, much less transcendental; although ...
- 47.4k
18
votes
Accepted
Extending Apéry's proof to Catalan's constant?
Summary:
The continued fraction, the recurrence and the explicit form of the sequence are interchangeable and for the Apéry numbers, we don't know what come first. This extend to other constructions ...
18
votes
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
It is straightforward to show that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. Indeed, if this equals a rational number $r\in\mathbb{Q}$, then $2\sin(2\pi r)=\sqrt{4/3}$. However, $2\sin(2\pi r)$ ...
- 105k
17
votes
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
As Frits Beukers writes in http://www.staff.science.uu.nl/~beuke106/caen.pdf "Ironically all generalisations tried so far did not give any new interesting results. Only through a combination of ...
- 16.5k
15
votes
Are rationals everywhere equally dense?
Rationals are equidistributed in the sense that If you take any "nice" function, then if you approximate the integral of $f$ over (say) $[0, 1]$ by $\frac{1}{N_B} \sum f(r),$ where $N_B$ is the number ...
- 96.4k
15
votes
Accepted
Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?
The irrationality of $\log \pi$ is an open problem (see for example this recent paper).
It is expected to be transcendental (page 34 of this slides by Michel Waldschmidt), and in fact this follows ...
- 17.6k
14
votes
Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
This is studied in detail in Sums of Fractional Parts of Integer Multiples of an Irrational
The
sum
$$C(N)=\sum_{n=1}^{N} \bigl( \{ n \xi \} - \tfrac{1}{2}\bigr)$$
satisfies $|C(N)|>c\log N$ for ...
- 188k
13
votes
Accepted
Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
A counterexample for $n=2$ is the map $\varphi(x,y) = (x,(x^2-2)y)$. Each point $(r,s)\in\mathbb{Q}^2$ is the image of $\left(r,\frac{s}{r^2-2}\right)\in\mathbb{Q}^2$, but e.g. $(\sqrt{2},1)\not\in\...
- 9,061
12
votes
Irrationality measure of arctan(1/3)
The irrationality measure of $\arctan(1/3)$ is not known. It lies between $2$ and $6.096755\dots$. The lower bound is trivial (it holds for every irrational number), while the upper bound is the main ...
- 105k
12
votes
Fractional part power
For $x \in \mathbb{R}_{\ge 1}$ let $\phi(x) = (\lfloor x^n \rfloor + \frac{1}{2})^{\frac{1}{n}}$, where $n$ is the smallest positive integer such that $(\lfloor x^n \rfloor + \frac{1}{2})^{\frac{1}{n}}...
- 1,531
11
votes
Accepted
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?
Since $a,b$ are incommensurable, $(ax,bx)$ is asymptotically equidistributed
in the torus $({\bf R} / 2\pi{\bf Z})^2$.
[One proof is via a continuous version of Weyl's equidistribution criterion:
for ...
- 79.9k
11
votes
Accepted
Fractional part power
The OP asks for an instance of what Dubickas [1] has called a ${\cal Z}$-number: A real number $x>1$ for which there exists a real $\xi\neq 0$ such that $\{\xi x^n\}<1/2$ for every integer $n$.
...
- 188k
10
votes
Accepted
Is there a real valued function whose limit exists only on irrational numbers?
Arrange rationals in a sequence $q_n$, and set
$$f(x) = \sum_{n = 1}^\infty 2^{-n} \mathbb{1}_{[q_n,\infty)}(x),$$
where
$$\mathbb{1}_{[q_n,\infty)}(x) = \begin{cases} 1 & \text{if $x \geqslant ...
- 17.2k
10
votes
Accepted
Algebraic and rational parts of a real number
Let $\alpha$ be an irrational. We shall consider its continued fraction $[a_0;a_1,a_2,\dots]$. Recall some basic results about convergents of continued fractions (see e.g. here): letting $p_n,q_n$ be ...
- 28.2k
10
votes
Accepted
The square root of natural number expressed by an infinite series
So, here we have $P=2$ and $Q=1-m$.
Notice that
$$\frac{Q^n}{U_n(P,Q)U_{n+1}(P,Q)} = \frac{U_{n+1}(P,Q)}{U_n(P,Q)}-\frac{U_{n+2}(P,Q)}{U_{n+1}(P,Q)}.$$
By telescoping, it follows that
$$\sum_{n=1}^k \...
- 34.3k
8
votes
Irrationality of generalized continued fractions
There are such examples in Ramanujan's Notebooks, Part 2, page 116
- 5,624
8
votes
Irrationality measure of arctan(1/3)
As mentioned by GH from MO, the irrationality measure
for almost all real numbers is 2. However,
computing it for a particular number
is a notoriously difficult problem. For an irrational algebraic ...
- 6,891
8
votes
Accepted
Irrationality of $e^{x/y}$
I think this might be a solution.
The Continued Fraction Expansion of the hyperbolic tanh function discovered by Gauss is
$$\tanh z = \frac{z}{1 + \frac{z^2}{3 + \frac{z^2}{5 + \frac{z^2}{...}}}} \\\\$...
- 177
8
votes
Accepted
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
The length $R_N$ of the longest run in the first $N$ digits satisfies $R_N/\log_b(N) \to 1$ almost surely as $N \to \infty$, as first proved by Renyi, see the discussion in [1].
(Many references focus ...
- 14.2k
8
votes
Compilation of strategies to show that some constant is irrational
Another technique, which is discussed in the Mathoverflow question Irrationality proof technique: no factorial in the denominator is to show that $n!$ cannot be the denominator for any $n$. In ...
- 6,969
8
votes
Compilation of strategies to show that some constant is irrational
Don't forget the obvious: If the digits of $\alpha$ in any base are not eventually periodic, then $\alpha$ is irrational.
Mel Nathanson and I (Integers 14 (2014), A40, pp. 1--11) used this to prove ...
- 9,746
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