Questions tagged [transcendental-numbers]
Transcendental numbers are numbers that cannot be the root of a nonzero polynomial with rational coefficients (i.e., not an algebraic number). Examples of such numbers are $\pi$ and $e$.
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Power series where the number $e$ is a root
I have been going at this question for weeks now and couldn't find anything.
Can we have a series of the form:
$$f(x)=\sum_{n=0}^{\infty} a_n x^n$$
where $a_n$ are rationals and not all $0$ such that $...
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Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?
As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
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Is $\sqrt{2}$ an element of the set $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$? [closed]
I'm exploring the properties of the set formed by taking the modulo $\pi$ of natural numbers, specifically $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$. This set includes all values $k - 2n\pi$ where $0 \...
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The connection between $\pi$, $e$ and $20$ [closed]
It's well documented that $e^{\pi} \approx 20+\pi$. This can be explained using the following series:
$$\sum\limits_{k=1}^{\infty}\frac{8\pi k^{2}-2}{e^{\pi k^{2}}} = 1$$
The series is quickly ...
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We know existence of Transcendental raised to Algebraic Irrational equals rational, but what about opposite?
Introduction:
If we take $a=2^\sqrt[3]{2}$ which is transcendental by Gelfond-Schneider Theorem, and $b=\sqrt[3]{4}$ which is algebraic irrational because it is root of monic-irreducible polynomial ...
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If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$
If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$
Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
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Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$?
Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$? $i \in \mathbb{Q}(e+i)$? Intuition tells me $i$ is not contained in $\mathbb{Q}(e+i)$ because it should somehow contradict the fact that $e$ ...
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Are the vast majority of irrational numbers, transcendental? [duplicate]
It is often stated that the vast majority of real numbers are irrational. Does it also follow that the vast majority of irrational numbers are transcendental?
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Constructing number between zero and one by concatenating digits from square root of primes
Let $a_{n}=$ the first $p_{n}$th digits to the right of decimal point of the square root of the $n$th prime.
Example:
$\sqrt{2}=1.414213562...$
So, $a_{1}=41$
$\sqrt{3}=1.73205080...$
So, $a_{2}=732$
$...
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Is transcendental number to non zero algebraic power always a transcendental?
My motivation to this question is that we know $e^{a}$ is transcendental , where $a$ is Non-Zero-Algebraic, using Lindemann Theorem, but is it true for all transcendental numbers not only $e$?
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I want to ask for good references of linear algebra over rational numbers?
My current studies in algebraic number theory have led me to observe the frequent interplay between linear algebra concepts over the field of rational numbers. This connection becomes particularly ...
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A simple, concrete example of a transcendental element.
I am writing an article on transcendental numbers and I'm wondering if it is possible to construct a "simple" example of a transcendental element in a field extension.
When thinking about ...
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Formula including divisors sum ($\sigma$), Euler Gamma ($\gamma$), $\pi$ and $\ln \pi$ [closed]
An interesting formula arose during the investigation of divisors sum efficient calculation. Actually the below series converges very slowly, as every series containing $\gamma$ :)
$$\sum _{k=1}^{\...
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How do I check if this number is transcendental?
Two days ago, I tried to create an infinite series that might be able to generate a transcendental number, and when I checked the proper definition, it was mentioned that, it is a number that cannot ...
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Images of a vector under the Galois differential group span the solution set
I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
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