# Questions tagged [transcendental-number-theory]

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### Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...

**11**

votes

**1**answer

579 views

### Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...

**1**

vote

**0**answers

78 views

### unknown sequences of rational numbers with sum of a transcendental number [closed]

Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form :
$e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}...

**5**

votes

**1**answer

324 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 ...

**7**

votes

**1**answer

260 views

### Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$

Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...

**6**

votes

**2**answers

223 views

### Algebraic exponential values

Is there a non-zero real number $t$ for which there exist infinitely
many prime numbers $p$ with $p^{it}$ an algebraic integer?
I would even be surprised to find a real $t \neq 0$ with both $2^{it}$...

**0**

votes

**0**answers

40 views

### Equality of combinations of exponentials and logarithms

Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...

**0**

votes

**0**answers

25 views

### Additive and linear independence of squares of logarithms

Since Alan Baker work, a lot is known about linear combinations
of logarithms, and more. However, squares of logarithms can be
useful and attractive too.
Let $\ Q\ :=\ \{q_1\ \ldots\ q_n\}\ $ be ...

**4**

votes

**1**answer

225 views

### Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have
$$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$
or perhaps with the weaker estimate with the right side replaced ...

**5**

votes

**0**answers

345 views

### Why does Faltings' Siegel lemma imply Siegel lemma?

Recall the Siegel lemma:
Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...

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votes

**0**answers

72 views

### Closed form of a Fredholm number

It is known that the so called Fredholm number $\sum _{n=0} ^{\infty} \beta^{2^n}$ is transcendental for any algebraic $\beta$ with $0<|\beta|<1$. However, it may be the case, that a closed form ...

**5**

votes

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163 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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vote

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98 views

### Siegel lemma with one contrainst

Les $A=(a_{ij})_{\substack{1\le i\le m\\1\le j\le n}}$ ($n>m$) be a matrix of integers entries. Can one determine a "small" (depending on the size of entries of $A$) solution $(x_1,x_2,\cdots,x_n)\...

**4**

votes

**1**answer

316 views

### Existence of normal number except random numbers

For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...

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vote

**0**answers

128 views

### Is there any irrational algebraic number among the set? [closed]

Suppose $S$ is set of numbers such that every number in it expands in decimal digits,every digit is 0 or 1,and $\lim_{n\rightarrow\infty}\frac{C_{n}(0)}{n}=\frac{1}{2}$ where ${C_{n}(0)}$ and ${C_{n}(...

**7**

votes

**1**answer

748 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 ...

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votes

**1**answer

3k views

### Is 0.24681012141618202224… transcendental?

Is a number whose infinite decimal part is the sequence of even numbers, transcendental? How about a number whose infinite decimal part is the odd numbers? Would the odds be more difficult to prove ...

**-1**

votes

**1**answer

150 views

### Is there a fixed positive integer $n $ for which :$\log^n x+e^{nx}=1$ with $0<x <1$? [closed]

I have tried to know if there is a fixed positive integer $n $ and $x$ is a real number such that $0<x<1$
for which :$\log^n x+e^{nx}=1$ but i can't succeed ,
Then my question here is :
...

**3**

votes

**1**answer

571 views

### Chudnovsky algorithm and Pi precision

What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.

**3**

votes

**2**answers

221 views

### Integer programming of free energy

Given $k$, $f(x) = e^{x_1}+e^{x_1+x_2}+\cdots+e^{x_1+x_2+\cdots+x_k},$ $x=(x_1,x_2,\ldots,x_k),$ where $x_i \in \{0,1\}$.
We want to compute: $\inf_{x \neq y}|f(x)-f(y)|$ or a lower bound of $\inf_{x ...

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votes

**1**answer

443 views

### Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]

I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...

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votes

**1**answer

474 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

**0**answers

126 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 $r=r_1+r_2-1$ ...

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votes

**0**answers

236 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

73 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

**1**answer

209 views

### Algebraic independence of $a^a$ and $b^b$ for algebraic irrationals $a,b$

Let $a,b$ be algebraic irrationals.
Are there conjectures or unconditional results about the algebraic
independence of $a^a$ and $b^b$?
Probably Schanuel's conjecture is related,
maybe only $\log{a},...

**0**

votes

**2**answers

285 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 ...

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votes

**0**answers

545 views

### Doubts on a paper by Lang in Transcendental Number Theory

... namely, in the seventh page of his Transcendental numbers & diophantine approximations. I do hope that somebody in the site has read this very paper recently... Here is a screenshot of the ...

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votes

**1**answer

340 views

### 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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votes

**2**answers

779 views

### Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?

I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as ...

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votes

**0**answers

103 views

### Inhomogeneous approximation mod p

I am interested in the following discrete analog of Kronecker's inhomogeneous approximation theorem.
Given $x_1,\ldots,x_n$ distinct residue classes modulo a prime $p$, and further residue classes $...

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votes

**1**answer

290 views

### Linear independence of p-adic logarithms (analog of Baker's theorem)

We have the following theorem of Baker:
Theorem 1. Let $\alpha_1, \ldots, \alpha_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha_1, \ldots, \log \alpha_m$ are linearly ...

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votes

**2**answers

720 views

### Could there be a special-function counterexample to Schanuel's conjecture?

It is not too hard to show that if Schanuel's conjecture is true, then the only algebraic numbers admitting a "closed-form expression" (as defined precisely in this paper) involving $e$, $\pi$, and ...

**-1**

votes

**1**answer

300 views

### rational power transcendental [closed]

A common example that rational power transcendental number is rational is $2^{\log_2 3}$.The irrationality of $2^{e}$ and $2^{\pi}$ is still unknown. Gelfond-Schneider theorem gives the answers only ...

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votes

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279 views

### Effective Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem states that if $\alpha_1, ..., \alpha_n$ are algebraic numbers which are linearly independent over the rational numbers $ℚ$, then $e^{\alpha_1}, ..., e^{\alpha_n}$ ...

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votes

**2**answers

638 views

### A transcendence question involving the exponential function

Let $(z_n)$ be a sequence of complex numbers satisfying $|z_n|\to +\infty$ and such that $\{e^{z_n}\mid n \in \mathbb{N}\}$ is infinite.
Is it always true that $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ ...

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votes

**0**answers

270 views

### Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers.
Question 1: Is it true that $\mathbf{C}$ is ...

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votes

**0**answers

87 views

### Transcendence of a $q$-series

Let $q\ge 2$ be an integer. Fourier's proof of irrationality of $e$ adapts to prove the irrationality of
$$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$
Is this number knwon to be ...

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votes

**3**answers

1k views

### Unexpected applications of transcendental number theory?

In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of ...

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598 views

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...

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votes

**0**answers

154 views

### Transcendency of certain integrals

Let $p$ be an even monic polynomial with rational coefficients of degree at least $4$.
Can the integral
$$\int_{-\infty}^\infty e^{-p(x)}dx$$
be an algebraic number? Is anything known about ...

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votes

**1**answer

252 views

### Transcendence of a ratio of p-adic logarithms

Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that ...

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votes

**0**answers

114 views

### On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...

**5**

votes

**1**answer

432 views

### Transcendence of $\Gamma(1/3), \Gamma(1/4)$

This is a re-post from MSE as I did not get even a single comment there.
Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...

**32**

votes

**2**answers

1k views

### Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, I thought it was easy to show that there are no non-trivial integer solutions to this equation, but then I ...

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votes

**2**answers

599 views

### For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...

**11**

votes

**1**answer

445 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 ...

**5**

votes

**1**answer

212 views

### transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...

**3**

votes

**0**answers

163 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...

**8**

votes

**5**answers

1k views

### Small values of a polynomial evaluated at roots of unity

The MO answer https://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant $C(\...