Why is the expected value of this random nested radical equal to $4/\pi$?
To celebrate Pi Day, I took a look at some classic identities for $\pi$ again, particularly the nested roots associated with Viète’s formula. Such as this one:$$\pi=\lim_{k \to\infty} 2^k...
View ArticleA continued fraction for Baxter's four-coloring constant
I found the following infinite continued fraction:$$\operatorname{C_{B4CC}}=\cfrac{2}{1+\cfrac{2}{1+\cfrac{6}{1+\cfrac{3}{1+\cfrac{10}{1+\cfrac{4}{\ddots}}}}}}$$where $\operatorname{C_{B4CC}}$ denotes...
View ArticleDistribution of the digits of $\pi$
Can anything be stated about the distribution of the digits of $\pi$, i.e., if I were to sample $n$ digits of $\pi$, can anything be said about the probability to observe certain digits, or is there...
View ArticleA continued fraction for the reciprocal of Gauss's constant
I found the following infinite continued fraction:$$\frac{1}{G} = \frac1{(2\pi)^{-3/2}\,\Gamma^2(1/4)} = \cfrac{4}{1+\cfrac{5}{1+\cfrac{6}{1+\cfrac{9}{1+\cfrac{8}{1+\cfrac{13}{\ddots}}}}}}$$where $G$...
View ArticleIs it mathematically correct to write $e^{i\pi}=i^2$?
I know that Euler's identity gives $e^{i\pi}=-1$, and since $i^2=-1$, it follows that $e^{i\pi}=i^2$.My questions are:Is it mathematically correct to write $e^{i\pi}=i^2$, or is this considered poor...
View ArticleProving $\pi^3 \gt 31$
$$\large \pi^3 \gt 31$$Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand.It is extremely easy with the use of any calculator, so I was wondering now:Can...
View ArticleAbout Legendre's like relations generalizations.
ContextBeing:$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2{t}}}=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(2n)!^2k^{2n}}{2^{4n}n!^4},\hspace{.5cm}k...
View ArticleDoes every digit in base $n$ appear in the expansion of $\pi$? [duplicate]
So I was thinking of the conjecture that every finite combination of digits appears somewhere in the expansion of $\pi$. This led me to a wonder:For every integer base $n \ge 2$, does every digit...
View ArticleWhat is the probability of a specific sequence of 11 digits occurring in a...
This isn't homework. I'm actually wondering how likely it is that any particular 11-digit telephone number will occur in the first billion digits of $\pi$. My probability course was way too long ago,...
View ArticleOn Ramanujan's fastest series.
ContextWith some effort we can show that Ramanujan's fastest series implies:\begin{align}\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}\end{align}with:...
View ArticleProof of an Elliptic Integral Relation
In celebration of Pi Day, I messed around with the following Ramanujan formula:$$\sum_{n=0}^{\infty} \binom{2 n}{n}^3 \frac{42 n+5}{2^{12 n+4}} = \frac1{\pi} $$It turns out that, through some...
View ArticleApproximation of $\pi$ by approximating different distributions
I think I found an approximation for $\pi$. I know, there are like $10,000$ approx., but I gave it a try. I found the following expression:$$\pi \approx...
View ArticleHow does this telescoping infinite product linking $e$ and $\pi$ work? [closed]
I recently encountered a paper titled "A Masterclass in Pure Rational Geometry & Analytical Limits" (dated June 7, 2025). Section 4 of the document presents a striking identity that expresses $\pi$...
View ArticleIs this two-stage continued-fraction approximation just a disguised...
I would like to understand how to classify the following approximation to $\pi$.Define$$N(a)=\sqrt{\frac{\pi}{\sqrt{\frac{\pi}{\sqrt{\pi}a}}}}.$$A direct simplification...
View ArticleHow can this two-stage continued-fraction approximation to pi be classified?...
I am looking for references or known related constructions for the following elementary approximation to $\pi$. It arises from the repeating...
View ArticlePatterns in pi in "Contact"
In Carl Sagan's novel Contact, the main character (Ellie Arroway) is told by an alien that certain megastructures in the universe were created by an unknown advanced intelligence that left messages...
View ArticleEvaluating a Mellin-Barnes integral arising from a quadratic formula for...
While exploring the numerical closeness of $\frac{1}{\pi}$ and $1 - \ln 2$, I derived a quadratic whose positive root is $\pi$. The formula involves a constant $\mathcal{C}$ currently defined in terms...
View ArticleConjecture: $\sum_{j=1}^J\arcsin\left(r(j)\sqrt{C(j)}\right)\neq2\pi$, for...
Let $b$ be a positive integer. Let $r(b)$ be a rational number depending on $b$ and being nonnegative.Let $C(b)$ be a prime depending on $b$ and different for every $b$. [1]Consider for positive...
View ArticleA cool integral:...
I was looking at the expression $\ln(e^{x}-e^{-x})$ and found that the zero is at $x=\ln{\phi}$, where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I...
View Article![Cloonee, Yandel & ATLAC – Movimiento – Single [iTunes Plus M4A]](http://i.imgur.com/Q8tO40p.jpeg)
