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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 :
$${...
user99666
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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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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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