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6
votes
2answers
869 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 ...
19
votes
14answers
4k views

Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.) 1) (Trivial) ...
6
votes
3answers
699 views

English translation of Voronoi's dissertation

I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
9
votes
1answer
521 views

Poles from the Continued Fraction Expansion of the Tangent Function?

Consider the well known continued fraction expansion $$ z \tan z = \frac{z^2}{1 - \cfrac{z^2}{3- \cfrac{z^2}{5 - \ldots}}} $$ of the tangent function going back to Euler and Lambert (Lambert used it ...
9
votes
1answer
329 views

Lengths of continued fractions for the numbers with fixed ratio

Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and ...
22
votes
0answers
926 views

Ramanujan's $\tau(n)$ and continued fractions

In D.H. Lehmer's paper "Ramanujan's function $\tau(n)$, (Duke J. Math v. 10 1943, pp. 483-492), Lehmer states the Ramanujan conjecture $|\tau( p )|< 2p^{11/2}$, so that $p^{-11/2}\tau( p ...
6
votes
2answers
370 views

How Symmetric is Diophantine Approximation using Fractions with Square Denominators?

Let $S$ be an infinite set of positive integers. Let us say that a "best S-approximation" to a real irrational $r$ is a rational number $p/q$, with $p$ and $q$ integers and $q \in S$, such that for ...
7
votes
2answers
811 views

Simultaneous rational approximation of two reals using their continued fractions

Littlewood's well-known conjecture about simultaneous rational approximation is that for all $x, y \in \mathbb{R}$, $\liminf_{n \to \infty} n \Vert nx \Vert \Vert ny \Vert = 0$ (where $\Vert x \Vert$ ...
7
votes
3answers
1k views

distance formula in Farey graph?

Consider the usual "Farey graph", i.e. the 1-skeleton of the (essentially unique) triangulation of the disk by ideal triangles. If one insists that 1/0, 0/1, and 1/1 are vertices of a triangle then ...
2
votes
1answer
828 views

Periods of Continued Fractions

Does the period length $l(pq)$ of the continued fraction of $\sqrt{pq}$, for $p$ and $q$ primes, follow some type of divisibility property, say $$ l(pq) = c\frac{l(p)}{l(q)} \quad\text{or}\quad ...
5
votes
2answers
589 views

Lower bounds (or less) for the period of \sqrt(D) and related sequences.

This is a continuation of Lower bounds for period length of continued fraction of square root which is a continuation of Upper bound of period length of continued fraction representation of very ...
11
votes
5answers
1k views

Relation between indefinite quadratic forms and continued fractions

Let $D$ be a positive square free integer; for simplicity let's take $D$ to be $2$ or $3$ modulo $4$. Then ideal classes in $\mathbb{Z}[\sqrt{D}]$ are in bijection with matrices $\left( ...
1
vote
1answer
561 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 ...
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,...]?
2
votes
1answer
233 views

Is this related to the j-function?

I was browsing in Plouffe's inverter and found that $\frac{\exp(\pi)-\ln(3)}{\ln(2)}$ is very nearly $\frac{159}{5}.$ The continued fraction is [31, 1, 4, 12029125, ...]. Is this the same magic as ...
3
votes
0answers
186 views

Cassels' algorithm vs. “divided cells” algorithm

Cassels' algorithm mentioned in link text looks similar to Delone's “divided cells” algorithm. Are there any differences in these algorithms?
6
votes
0answers
978 views

Lower bounds for period length of continued fraction of square root

EDIT: Dror answered this to my complete satisfaction. The simple continued fraction for $$ \sqrt{n^2 + 1}$$ has length exactly the same, no dependence on $n$!!!!! No lower bound of the type I wanted ...
10
votes
1answer
539 views

Applications of pattern-free continued fractions

Questions about continued fractions reminded me about a related diophantine problem. I am not quite sure that diophantine equations are still in fashion but $$ 1^k+2^k+\dots+(m-1)^k=m^k, $$ the ...
12
votes
3answers
2k views

Upper bound of period length of continued fraction representation of very composite number square root

Given natural numbers of special very composite form, like primorials or factorials, how to give some useful upper bound limit of continued fraction period length of their square roots? I'm not a ...
22
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 ...
14
votes
2answers
1k views

Is there an elementary proof of a result about the parity of the period of the repeating block in the continued fraction expansion of square roots

It is a known fact that for a Prime $P$, $P\equiv 1$ mod $4$ iff the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd. I have an elementary proof ...
15
votes
8answers
1k views

Continued fractions using all natural integers

What can one say about the set of continued fractions $[0;a_1,a_2,\ldots]$, where $a_1,a_2,\ldots$ are a permutation of the set of natural numbers?
14
votes
5answers
998 views

Algorithm generalizing continued fractions for non-quadratic algebraic numbers

The continued fraction algorithm generates an integer sequence which terminates for a rational number, is periodic for the roots of irreducible integer quadratics, and is non-periodic for other ...