# 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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### Solving 2-dimensional recurrence matrix of homogenous polynomials

I know this isn't research-level mathematics, but I posted this on stack exchange, even offered a bounty but got no responses and no comments, so I am hoping to get an answer here.
In $2012$, ...

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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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### Special case of Schanuel conjecture

This is one of the simplest special cases of Schanuel conjecture.
For $f(x),g(x) \in \mathbb{Z}[x]$ define $F(x)=f(x) + i g(x)$.
Are there $f(x),g(x)$ and real $x_0$ such that $C=\exp{F(x_0)}$
is ...

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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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334 views

### 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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### 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 ...

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### Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...

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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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### 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 ...

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### Computing all “suboptimal” rational approximations to $\pi/2$

I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy
$$ n \epsilon(n)^2 \leq \tau $$
where $\tau$ is a known ...

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### Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...

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### 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},.....

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### Conjecture on irrational algebraic numbers

Conjecture:
For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
Questions:
Has this conjecture been ...

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### 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 ...

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### 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 $\...

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### 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\...

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### irrationality of the p-adic exponential

I would like to illustrate my lecture on p-adic numbers with some elementary results.
I proved that the series $e^p=\sum_{n\ge0}\frac{p^n}{n!}$ converges in $\mathbb Q_p$ for every prime $p$.
Now I ...

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### If $a$ is irrational, must $a^a$ be irrational? [closed]

It is known that $\sqrt{2}^{\sqrt{2}}$ is irrational. Is it true that for any irrational number $a$, $a^a$ must be irrational?

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### A question on subgroup-restricted irrationality measures

The irrationality measure μ(x) of a positive irrational number x is defined to be the supremum of the exponents e such that |x - p/q| < 1/q^e has an infinite number of solutions p/q. By the ...

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### Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...

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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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### $\mathbb{S}^2$ equivalent to frac$(n \alpha)$ equidistribution on $\mathbb{S}^1$

Let $\operatorname{frac}(x) = x - \lfloor x \rfloor$ be the fractional part of $x$.
Then, for $\alpha$ irrational, $\operatorname{frac}(n \alpha)$, $n=1,2,\ldots$, distributes
randomly in $[0,1)$, ...