# 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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### 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
$$\frac{\log a \log b}{\log c \log d} \notin Q ?$$
The theorems of ...

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### Prove that the power of irrational number with irrational exponent can be a rational number [closed]

I stumbled upon the problem described in the title and sketched the following:
Let $\mathbb{Q}' = \mathbb{R} \setminus \mathbb{Q}. $
Given $\{a,b,x,y\} \in \mathbb{R}_+, \alpha \in \mathbb{Q}$ let'...

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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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### What about a second Hardy–Littlewood conjecture for Beatty primes?

I'm curious to know about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia Second Hardy–Littlewood conjecture) is feasible for Beatty primes. As in the ...

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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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### Two exponents being algebraic

Schanuel conjecture implies this, so likely it is true.
Let $f(x),g(x)$ be polynomials with coefficient in $\mathbb{Z}[i]$.
Assume that for some complex number $x_0$, both $\exp{f(x_0)}$ and
$\exp{...

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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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### Combination of irrationals

Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...

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### Are rationals everywhere equally dense? [closed]

I would like to know is there any notion of density over the rationals with which we could determine are rationals everywhere equally dense on the real line, because, for example, I am not sure would ...

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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 ...

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### 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 ...

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### Why is the Euler-Mascheroni constant not a Liouville number?

Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known?
Any pointers to the literature are ...

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### “Transcendental tilings”: Do they exist?

Let $T$ be a tiling of the plane.
Fix an origin and shoot a ray $r$ from the origin.
Mark off points $p_i$ along $r$ separated by unit distance.
Compute from $r$ a binary number $0 < b(r) < 1$ ...

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### Irrationality of generalized continued fractions

An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$
is irrational. Now for a generalized continued fraction:
$$\frac{a_1}{b_1 + \frac{a_2}...

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### Chudnovsky algorithm and Pi precision

What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.

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### Is $x=\frac{1}{2}$ the solution of this equation $\zeta(2)= 1+{{{{x}^{x}}^{x}}^{x}}^{\cdots } $?

I would like to study the irrationality of ${{{{x}^{x}}^{x}}^{x}}^{\cdots } $
for $x=\frac{1}{2} $ using the irrationality of $\zeta(2)$ .
Some computations in wolfram alpha show to me that :
$${...

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### Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]

I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...

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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 ...

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### Researching the irrationality of a number [closed]

I am conducting a little research on checking if a number, written in positional numeral system is irrational.
Let $h^p_n$ be the most right non-zero digit of number $n!$ written in numeral system ...

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### 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 ...

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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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### 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 ...

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### 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 ...

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### Fractional parts of two multiples [duplicate]

There is a theorem (I can't remember its name) saying that for any irrational number $x$ and any $0<a<b<1$, there exists a positive integer $n$ such that $\{nx\}\in (a,b)$, where $\{\cdot\}$ ...

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### Is being rational decidable?

Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number ...

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### 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 ...

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### 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 ...

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### Approximating a real by a ratio of primes

Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$.
I seek smallest primes $p$ and $q$ such that
$$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$
Q. What upper bound $u(...

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737 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 ...

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### 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 ...

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### 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 ...

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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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### 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 ...

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### 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 ...

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### 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 ...