2
votes
1answer
153 views

New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime ...
6
votes
1answer
190 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
0
votes
1answer
41 views

What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
0
votes
0answers
88 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
5
votes
1answer
103 views

For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$?

Inspired by this question I ask this. For which $a$ is $n\lfloor a\rfloor+1\le \lfloor na\rfloor$ true for all sufficiently large $n$? The original question concerned $a=e$, the usual ...
6
votes
1answer
153 views

General Continued Fractions and Irrationality

A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is: If, in the continued fraction ...
4
votes
0answers
112 views

Finding a closed expression for a calculated value.

Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic ...
11
votes
3answers
436 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
3
votes
1answer
380 views

How to express an irrational as a continued fraction in computer with high precision?

Background I'm writing a C++ library for continued fraction using MPIR (Multiple Precision Integers and Rationals) library http://www.mpir.org/ due to the limitation of built-in ...
4
votes
4answers
272 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, e and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
8
votes
3answers
711 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...