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22
votes
0answers
422 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 ...
14
votes
0answers
361 views

Integer-valued power towers

$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$ Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$. Are there any positive ...
11
votes
0answers
520 views

Connection between Infinite continued fractions, elliptic integrals and AGM

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + ...
10
votes
0answers
501 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 ...
5
votes
0answers
100 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 ...
4
votes
0answers
306 views

A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
3
votes
0answers
129 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}= ...
2
votes
0answers
182 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 ...
2
votes
0answers
72 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 ...
2
votes
0answers
114 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 ...
2
votes
0answers
254 views

schanuel's conjecture and real root of $x+e^x=0$

Schanuel's conjecture states: -If $\alpha_1,\alpha_2,...,\alpha_n$ are complex numbers linearly independent over $\mathbb{Q}$ then the trascendence degree of the field ...
0
votes
0answers
133 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
0
votes
0answers
176 views

On the irrationality measure of generalized Stoneham numbers

Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and ...