# Questions tagged [transcendence]

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### Transcendence of Euler series

Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
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### On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio

Define the real number $\xi$ satisfying $$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$ where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
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### Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
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### If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?

This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$, ...
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### A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero

Let $f,g \in \mathbb{C}[x,y]$. There is a well-known result, that can be found for example here, pages 19-20, that says the following: $f,g$ are algebraically dependent over $\mathbb{C}$ if and ...
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### Generalizations of Lüroth theorem

Let $k$ be an arbitrary field (I do not mind to take $k=\mathbb{C}$, if things are easier in this case). A more general version of Lüroth theorem says that a field $L$, $k \subset L \subset k(x,y)$, ...
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### Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...
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### Nielsen--Schreier for fields

Is it true that a subextension of a purely transcendental extension is itself purely transcendental? In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
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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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### Solving a transcendental equation, in closed form

There is a change of variable between two varibles $q$ and $Q$ as the following: $$q=Q\exp(2f(Q))\quad\quad \quad (*)$$ where $f(Q)$ is given by $$f(Q)=\sum_{d=1}^\infty \frac{(2d-1)!}{(d!)^2}Q^d$$ ...
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### Transcendence of $e^{\frac{\pi^2}{12 \log 2}}$

Is it known whether $e^{\frac{\pi^2}{12 \log 2}}$ is transcendental or algebraic? This number showed up in this other question.
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### $\mathbb{Z}$-linear independence of arguments of units in non-CM number fields

Playing a little bit with Groessencharacters a stumbled on the following question: Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$ ...
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### Numbers with known finite irrationality measure greater than 2

For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that  \left| \alpha-\frac{p}{q}\right|...
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### Random Sequence : Definition of [closed]

"A sequence of bits is random if there exists no Program shorter than it which can produce the same sequence." ~ Kolmogorov Q: How do the digits of Pi fall as a random sequence based on the above ...
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### Why is it hard to prove that the Euler Mascheroni constant is irrational?

Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?
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### What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...
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### quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N$, when written as regular continued fractions (R.C.F.), yield what can be called a ...
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### Transcendental numbers that are “suspected” to be algebraically dependent without conjectured relation?

I am experimenting with a solver for finding algebraic dependencies and would like to test it on more data sets. Are there transcendental numbers that are "suspected" to be algebraically dependent ...
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### Is this seemingly novel irrational constant also transcendental?

I recently discovered a constant that is constructed as follows: $\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$ Furthermore I can prove that it is an irrational number ...
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