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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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 ...
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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 ...
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Show that there are infinitely many well-separated grids from a fixed set of points
I have stumbled upon a question which naturally arises when trying to bin a set of $n$ points into equispaced bins such that they are sufficiently well separated from the bin edges.
Take $n$ points $...
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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 ...
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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 ...
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Two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number
Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?
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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
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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 ...
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Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$
Consider the series
$$
\sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2})
$$
where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
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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 ...
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Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$
In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
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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 ...
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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
...
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Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$
This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
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Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
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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(...
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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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Can you identify this irrational number?
There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
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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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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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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 $a/b=\sqrt{2}$ ...
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Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
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Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?
I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
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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 ...
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Help with this irrationality proof
I have a real number, that is quite messy so I'll just call it $x$. I want to prove it's irrational. It's a proof by contradiction. The contradiction will rise if I assume $x$ is a rational number $p/...
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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 ...
user215601
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Are mantissas of irrationals provably unique, at a given precision? [closed]
Many thanks to all responders!
Is there any research as to the uniqueness of mantissas of irrationals? It's easy to see that the mantissa of the square root of 5 (0.236067977...) and the mantissa of ...
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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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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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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 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,
$$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...
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Irrationality of generalized continued fractions
An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z\setminus\left\{0\right\})$$
is irrational. Now for a generalized continued fraction:
$$\...
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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 ...
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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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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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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:
$$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...
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Multiplying by irrational numbers in combinatorial problems
This is getting no attention on stackexchange.
Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$.
It had escaped my attention until last week, ...
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Irrationality of this trigonometric function
I'd like to prove the following conjecture.
Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$).
Then
$f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$
is irrational if ...
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Is there a real valued function whose limit exists only on irrational numbers?
I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational.
I tried variations ...
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Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]
I came across this apparent random question in some math questions website. At first, I thought it was easy to show that there are no non-trivial integer solutions to this equation, but then I ...
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Irrationality of $e^{x/y}$
How to prove the following continued fraction of $e^{x/y}$
$${\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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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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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'...
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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
$$\sum_{n=1}^\infty \frac{(2+\...
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Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
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Elementary proof of the equidistribution theorem
I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...
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Upper bounds on the irrationality measure of the arctan of an algebraic number
Let $x$ be an algebraic number. Must $\arctan(x)/\pi$ have finite irrationality measure? Are there any useful upper bounds?
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Irrational rotations are rank 2 by intervals without spacers
Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$.
S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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Irrational combination of rationally independent polynomials
Let $p_1,\dotsc,p_k: \mathbb{N} \to \mathbb{Z}$ be rationally independent polynomials with zero constant term. If $t_1,\dotsc,t_k \in [0,1)$ are not all rational, is it true that the polynomial
$$p(n)...
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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, ...
