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14
votes
1answer
959 views

Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because ...
23
votes
4answers
2k views

How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
17
votes
6answers
3k 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,...]?
9
votes
1answer
1k views

The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's zeta function can be expressed as a continued fraction as follows \begin{align*} \zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty ...
8
votes
2answers
714 views

Length of Hirzebruch continued fractions

Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
8
votes
1answer
286 views

Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
6
votes
2answers
876 views

Applications of periodic continued fractions

Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question. What can you add to the following ...
4
votes
1answer
787 views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...
5
votes
1answer
288 views

Algorithm to determine sign of a polynomial

I've been working with a collaborator (Arek Goetz) on a dynamics problem involving piecewise isometries (a map $T$ on a domain $X$ (say a subset of the plane) such that $X$ is divided into a finite ...
3
votes
0answers
238 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 ...
1
vote
1answer
572 views

Is it possile for all real algebraic numbers to have continued fractions with bounded partial quotients ?

Roth's theorem states that for every real algebraic $\alpha$ and $\epsilon>0$, there is a $c>0$ such that $$|\alpha -\frac{p}{q}| > \frac{c}{q^{2+\epsilon}}.$$ Lang's conjecture strengthened ...