Questions tagged [irrational-numbers]
An irrational number is a real number that cannot be expressed in the form $\frac{n}{m}$ where $n$ and $m$ are integers.
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Irrational numbers [closed]
1.There exist $\alpha_1,..., \alpha_n$ irrational numbers greater that $1$, such that
$\{\alpha_1^k\}+...+\{\alpha_n^k\} \ge cn$ for $c>0$ and $n, k$ positive integers sufficiently large
2.there ...
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Irrational combination of rationally independent polynomials
Let $p_1,\dotsc,p_k: \mathbb{N} \to \mathbb{Z}$ be rationally independent polynomials with zero constant term. If $t_1,\dotsc,t_k \in [0,1)$ are not all rational, is it true that the polynomial
$$p(n)...
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Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$
This is inspired by this Math.SE question, for $a=1$.
Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series
$$\sum_{n=1}^\infty \frac{(2+\...
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What is known about constructively irrational numbers?
Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...
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The power of irrationality: ${\rm sin}(x) + {\rm sin}(\pi x)$ [closed]
I'm a physicist studying undulatory phenomena. Reducing the problem, I find the issue lies on the relative irrationality of the angular frequencies of two superimposed waves. To state it simply, ...
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System of two delayed functional equations with dilatations
Consider the operator $S:L^{2}(0,s+r)\longrightarrow L^{2}(0,1)^{2}$ defined
by
\begin{equation*}
Su(x)=\left( au(xs)+bu(xs+r),cu(xr)+du(xr+s)\right) ,
\end{equation*}
where $a,b,c,d$ are positive ...
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Irrational rotations are rank 2 by intervals without spacers
Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$.
S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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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?
$$\begin{equation*} \zeta (3)=...
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Irrationality of this trigonometric function
I'd like to prove the following conjecture.
Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$).
Then
$f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$
is irrational if ...
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Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function.
If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
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The square root of natural number expressed by an infinite series
Can you prove or disprove the following claim:
Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then,
$$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...
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The constant $\pi$ expressed by an infinite series
I am looking for the proof of the following claim:
First, define the function $\operatorname{sgn_1}(n)$ as follows:
$$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...
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Ergodic Theory and Euler-Mascheroni Constant
I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a ...
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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 ...
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On the degree of irrationality of two irrational numbers and their rational (in)dependence
Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ?
ADDENDUM: What if $...
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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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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?
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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 ...
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Rational linear subspace corresponding to an irrational vector
Given a vector $v = (v_1, \ldots, v_n) \in \mathbb{R}^n$, we can associate a rational linear subspace with this vector: assume $\{1, v_i \text{ for }i \in I\}$ is a linear basis of $\{1, v_1, \ldots, ...
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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 ...
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Irrationality of $e^{x/y}$
How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...
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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 ...
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Looking for a proof that $\sqrt2 + \sqrt5 + \sqrt[3]3$ is irrational [closed]
What is the easiest way to prove that $\sqrt2+\sqrt5+\sqrt[3]3$ is irrational?
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Illuminating a just-barely irrational polygon
As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Corollary 3. Let $P$ be a rational polygon....
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When does the set of possible walk lengths start being $\varepsilon$-dense?
Let $\Gamma$ be a finite directed graph, and suppose each directed edge $e \colon a \to b$ has a positive real length. Suppose given vertices $x, y \in \Gamma$, and suppose there are infinitely many ...
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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, ...
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Is there a real valued function whose limit exists only on irrational numbers?
I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational.
I tried variations ...
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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 ...
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Is the super square root of $2$ irrational? [closed]
The super square root of $n$ is the solution/solutions to $x^x=n$. Is the super square root of $2$ irrational?
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Mapping $\mathbb P$ onto $\mathbb Q ^\omega$
Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...
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What numbers (irrational) in radicals are expressible as trigonometric rational fraction with only rational multiplies of $\pi$?
What irrational expressions $A$ with radicals can be expressed as trigonometric rational fraction (not a series) with only rational multiplies of $\pi$.
Example:
$ \frac{1}{\sqrt5} = \frac{\sin\frac{\...
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Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function
In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
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What is known about the irrationality of ratios and products of logarithms of integers?
Let $a,b,c,d$ be positive natural numbers such that $\{a,b\} \neq \{c,d\}$ and such that none are perfect powers. Is it true that
$$\frac{\log a \log b}{\log c \log d} \notin \mathbb{Q} ?$$
The ...
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Question about proof of irrationality of $\zeta(3)$ [closed]
I'm reading this article of Henri Cohen about Apery's proof of the irrationality of $\zeta(3)$ but I don't really get the details of "THEOREME 1".
My first doubt is about the relation $a_n \sim A \...
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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: ...
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Quotients of the irrationals
Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
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Possible Birkhoff spectra for irrational rotations
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
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O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval
O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval.
Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$,
$$E(n,\theta, I) ={ ...
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Rational Peano curves
An rr function (i.e. rational rational function) is a quotient
$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$
such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$
QUESTION Do there exist ...
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If $x^x=2$ then is $x$ expressible using elementary functions?
I have a curious question. Let $x∈\mathbb{R}^+$ such that $x^x=2$. I am aware that the Gelfond–Schneider theorem implies that $x$ cannot be algebraic. However, is it still possible that $x$ can be ...
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The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd
This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here.
We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
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Real number which is different from all rationals [closed]
By diagonalization, it is possible to construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$, there exists an index $i \in \mathbb{N}$ such that $r_i \neq q_i$ (where $x_i$ ...
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Subsets of particular values of $\zeta'(k)$ that contain irrational numbers
We consider the set of elements $\zeta'(2),\zeta'(3),\zeta'(4),\zeta'(5),\ldots$ where $\zeta(z)$ is the Riemann zeta function and $\zeta'(z)=\frac{d}{dz}\zeta(z)$ its derivative. Thus we consider ...
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Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$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'...
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Examples for a Golomb's result, and rationals as $\sum_{n\geq 1}\frac{|G_n|}{P(n)}$, where $G_n$ are Gregory coefficients and $P(x)$ a polynomial
After I was stuying the first pages of a chapter of the book [1], in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
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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'...
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infinite set of mutually irrational numbers which odd linear combinations approximate 0 badly
I'm looking for a set of real numbers $\{\lambda_i;i\geq 1\}$ such that for each odd $n$, one can control $\delta_n:=\inf| \sum_i \pm n_i \lambda_i|$ where the $n_i$ are natural integers that sum to $...
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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 ...
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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 $\zeta(3)$, ...
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$\psi(2,1/6),\psi(4,1/6)$ 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 $\...
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