# Questions tagged [transcendence]

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### Proof that pi is transcendental that doesn't use the infinitude of primes

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...
11k 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?
4k 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 ...
5k 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 ...
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 ...
4k views

### Showing e is transcendental using its continued fraction expansion

Can the transcendence of e be shown using its continued fraction expansion e = [2;1,2,1,1,4,1,1,6,...]?
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 ...
1k views

296 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 $L$, $k \subset L \subset k(x,y)$, ...
117 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 $L/K$ ...
324 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 ...
322 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?
391 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 ...
137 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 $a_i>0$ ...
84 views

### Transcendence of Euler series

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