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### 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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### 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 ...
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. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{... 0answers 504 views ### Connections of results in Harmonic analysis in the theory of Transcendental Numbers An entire function f is said to be of exponential type if there exist constants c and k such that |f(z)|\leq c e^{k |z|}. A famous result of Polya says if f is an entire function of ... 0answers 101 views ### On existence of rapid Arithmetic geometric procedure? We know that \pi can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of 2^n bits of \pi at nth ... 0answers 308 views ### A question on M. Mignotte's Paper: “Petho's Cubics” I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ... 0answers 131 views ### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers \sqrt{A} and the integer A What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,\sqrt{2} = [1;2,2,2,2,…]; while \sqrt{14}= [3;1,2,... 0answers 188 views ### Automorphism group of \mathbf{C} over \overline{\mathbf{Q}} Choose an embedding \overline{\mathbf{Q}}\rightarrow \mathbf{C} from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that \mathbf{C} is ... 0answers 75 views ### Transcendence of a q-series Let q\ge 2 be an integer. Fourier's proof of irrationality of e adapts to prove the irrationality of$$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$Is this number knwon to be ... 0answers 117 views ### Transcendency of certain integrals Let p be an even monic polynomial with rational coefficients of degree at least 4. Can the integral$$\int_{-\infty}^\infty e^{-p(x)}dx$$be an algebraic number? Is anything known about ... 0answers 259 views ### schanuel's conjecture and real root of x+e^x=0 Schanuel's conjecture states: -If \alpha_1,\alpha_2,...,\alpha_n are complex numbers linearly independent over \mathbb{Q} then the trascendence degree of the field \mathbb{Q}(\alpha_1,e^{\... 0answers 134 views ### Solution to system of polynomial equations Suppose A is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,P_2(x,y_1,\dots,y_n)=0,\vdotsP_k(x,y_1,\dots,y_n)=0 is a system of equations with coefficients over $\mathbb{Z}$, and ...
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 \$\gcd(a,...