# Questions tagged [transcendence]

The transcendence tag has no usage guidance.

54
questions

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votes

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45 views

### Lower and upper bound from transcendental equation

We have encountered the (in)equation
$$\gamma^\tau - (\gamma L_f)^j \left( \frac 1 {L_f} \right) ^\tau - \frac {1 - (\gamma L_f)^j} M = 0$$
(we need to solve it both for $<$ and $>$, but we are ...

**0**

votes

**2**answers

239 views

### Transcendence on $ \alpha+f(\alpha), \alpha f(\alpha) $ and $ \alpha/f(\alpha) $ where $ \alpha$ is transcendental

Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\...

**13**

votes

**2**answers

988 views

### Why is it easier to prove $e$ is transcendental than $\pi$?

Why is it easier to prove $e$ is transcendental than $\pi$?
I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...

**4**

votes

**0**answers

32 views

### Transcendence of values of Fredholm series at algebraic arguments

Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...

**29**

votes

**1**answer

931 views

### How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?

I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...

**3**

votes

**1**answer

182 views

### Can one define a degree of a period?

In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and assuming the zeta values to be ...

**5**

votes

**1**answer

272 views

### Irrationality of $e^{x/y}$

How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...

**0**

votes

**1**answer

140 views

### How do I get an analytical solution to this nonlinear equation?

I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...

**0**

votes

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73 views

### Transcendence à la Liouville

Let $\alpha\in\overline{\mathbb Q}^*$. How to prove that for a non ultimately constant sequence $(\varepsilon_n)_n$ with $\varepsilon_n\in\{0,1\}$, the number $\sum_{n\ge0}\frac{\varepsilon_n}{\alpha^{...

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95 views

### Transcendence of Euler series

Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?

**2**

votes

**1**answer

108 views

### 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 ...

**3**

votes

**0**answers

114 views

### 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 $W(1)=\Omega$ ...

**9**

votes

**1**answer

247 views

### Concerning $k \subset L \subset k(x,y)$

The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset ...

**0**

votes

**1**answer

324 views

### $a^b=b^a$ and algebraicity [closed]

Suppose $a$ and $b$ are reals such that $a^b=b^a$. If $a$ is algebraic, is $b$ algebraic too?

**6**

votes

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294 views

### $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...

**5**

votes

**1**answer

418 views

### Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...

**5**

votes

**0**answers

246 views

### Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...

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votes

**1**answer

234 views

### 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 $f,g \in k[h]$,
...

**5**

votes

**2**answers

203 views

### 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 ...

**2**

votes

**0**answers

334 views

### 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 $L$, $k \subset L \subset k(x,y)$, ...

**6**

votes

**1**answer

319 views

### 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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119 views

### 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 $L/K$ ...

**8**

votes

**1**answer

924 views

### 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 ...

**11**

votes

**1**answer

447 views

### "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 $0 < b(r) < 1$ ...

**2**

votes

**1**answer

266 views

### 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
$$f(Q)=\sum_{d=1}^\infty \frac{(2d-1)!}{(d!)^2}Q^d$$
...

**6**

votes

**1**answer

535 views

### 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.

**3**

votes

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135 views

### $\mathbb{Z}$-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 $r=r_1+r_2-1$ ...

**8**

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297 views

### Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.
Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...

**-1**

votes

**1**answer

75 views

### Preservation of algebraically dependence for derivative [closed]

It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ ...

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votes

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478 views

### algebraic independence of exponential functions

Let $a_1,\ldots,a_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)?
I ...

**2**

votes

**1**answer

327 views

### Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove a result which is used in A. Macintyre and A. J. Wilkie (1995), 'On the decidability of the real exponential field', in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI, p. ...

**0**

votes

**1**answer

142 views

### Transcendence of solutions of $\sum_{i=1}^n a_i b_i^x = 1$

I am currently doing things related to the Akra–Bazzi theorem. One element in that theorem is the following:
For $n>0$ and sequences of real numbers $a_i, b_i$ of length $n$, where all $a_i>0$ ...

**11**

votes

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481 views

### Transcendence of products of certain real algebraic numbers

Let
\begin{equation}
z := \prod_p p^{1/p^2},
\end{equation}
where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...

**4**

votes

**1**answer

244 views

### Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals.
Anyway, let $E$ be the "constructible numbers," ...

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votes

**3**answers

370 views

### Algebraically Independent Numbers and Affine Linear Maps

Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,...

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1k views

### 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|...

**-2**

votes

**1**answer

409 views

### 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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13k views

### 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?

**5**

votes

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1k views

### 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 ...

**3**

votes

**0**answers

478 views

### 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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301 views

### 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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**1**answer

2k views

### 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 ...

**4**

votes

**1**answer

359 views

### Finding purely transcendental parts of field extensions

If we have a field $K$ such that $K\cong K(t)$ (i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield $L\lt K$ such that $L\ncong L(t)$ and $...

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597 views

### Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.
A famous result of Polya says if $f$ is an entire function of ...

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**6**answers

1k views

### Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?

Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions ...

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999 views

### Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?

Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?
Is there a survey with up-to-date transcendence results?

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vote

**0**answers

328 views

### Extending transcendental numbers to multivariate polynomials [closed]

Transcendental numbers are a well-known phenomenon: a number $x$ is transcendental if no polynomial with integer coefficients has $x$ as its root, or $p(x)\neq0$ for all polynomials $p$ with integer ...

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votes

**6**answers

5k views

### What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...

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2k views

### Have all numbers with "sufficiently many zeros" been proven transcendental?

Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this ...

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**3**answers

6k views

### Work on independence of pi and e

It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.
What are some of the important results leading toward proving this?
What are the most promising ...