# Questions tagged [experimental-mathematics]

The utilization of advanced computing technology in mathematical research: new mathematical results discovered partly or entirely with the aid of computer-based tools.

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### integral transform integration over finite interval

please anyone help me to find the value of this integral $\int_{0}^{2 \pi} \frac{ (\omega)^{u-m}}{(\omega - \theta)^{t}} d\omega$ = ? where $u, m, t \in \mathbb N_{0}.$
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### Is probability perspective dependent?

Suppose there be a person 'A' doing coin unbiased coin flips and he gets 4 heads in a row, then the chance of the next being a tail is 97%, which is very very likely. But if person 'B' unaware of 'A'...
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### Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Recall that the Thue–Morse sequence$^{}$$\!^{}$$\!^{}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its ...
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### On a conjecture about the arithmetic function that counts the number of twin primes

In this post we denote (for a fixed positive integer or real number $x$) as $$\pi_2(x)=\#\{\text{ primes }p\leq x\,:\,p+2\text{ is also a prime}\}$$ the arithmetic function that counts the number of ...
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### On conjectures about the arithmetic function that counts the number of Sophie Germain primes

In this post we denote (for a fixed positive integer or real number $x$) as $$\operatorname{Germain}(x)=\#\{\text{ primes }p\leq x\,|\,2p+1\text{ is also prime}\}$$ the arithmetic function that counts ...
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### Sign of a function that involves the arithmetic function that counts the number of Ramanujan primes less than a given integer

In this post we denote the arithmetic function that counts the number of Ramanujan $R$ primes less than $x$ as $$\pi_{_{R}}(x)=\#\{\text{Ramanujan primes }R\text{ such that }R\leq x\}$$ thus is the ...
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### Probability for heads from an experiment

You are given a (perhaps biased) coin with probability for heads $p$ for some unknown $p$. Construct a random experiment such that with probability at least $0.99$ you can determine the value of $p$ ...
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### Point picking in a 1x1 square: probability of line segments connecting 2 random interior points to catacorner vertices intersecting in the square.

Like many of the best problems I found this one on twitter. "In a square with side length 1, two random points in the square are connected by segments to two opposite vertices. How likely is it that ...
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### A conjectured continued fraction involving non-polynomial patterns

After having been devoting some time for many years to experimental mathematics, I am thinking to publish the details of some of my most fruitful computing workflows for discovering identities. I ...
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### Prove that $\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$

Prove $$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$ Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my ...
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Using empirical methods, I conjectured that$^{}$$\!^{}$$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{... 1answer 156 views ### Semiperimeter of isosceles Heronian triangles. A Heronian triangle is a triangle with integer sides and area, named after Heron's formula which states that the area of a triangle with sides$a$,$b$, and$c$is $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ ... 1answer 16 views ### Understanding “Comparison” method in statistics I am struggling a lot with statistics so I decided to try David Freedman's Statistics book. In the book, first chapter, there is this explanation: A controlled experiment to show the vaccine was ... 1answer 306 views ### On binomial sums$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$and log sine integrals Seven years ago, I asked about closed-forms for the binomial sum $$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$ Some alternative results have been made. Up to a certain$k$, it seems it can be ... 0answers 214 views ### The number$\pi$in an unexpected context [This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square ... 0answers 283 views ### Figures and Numbers: Relating properties of geometric shapes and their Fourier series Consider two types of parametrized curves$\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$open curves$\gamma_\sim(t) = (t,a(t) + b(t))$closed curves$\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$... 0answers 200 views ### Digits of$\pi$in other bases? Pi day got me interested in thinking about whether$\pi$is a normal number - one whose digits are uniformly distributed no matter what base the number is written in. The OEIS sequence for$\pi$in ... 1answer 109 views ### Modified Pascal's triangle In Pascal's triangle, each number is the sum of the two numbers directly above it. I experimented with a modified Pascal's triangle. First a n-tuple of natural numbers is put to the nth row of the ... 0answers 444 views ### What is probability that: (a)a non leap year has 53 Sundays. (b)a leap year has 53 fridays. This one really bowled me over. I don't understand from where I should begin. I actually thought that a non leap year will have 52 weeks and 1 day. But I couldn't think how to find the probability . I ... 0answers 62 views ### Hubs in high-dimensional point clouds (distribution of “being within nearest neighbours of how_many other points”) The hubness problem is a phenomenon mentioned in this paper, for instance. It claims that for random data point clouds in high-dimensional spaces, there will be "many" hubs -- that is, points that are ... 0answers 75 views ### Let$x(n)=$product of first$n$primes.How often is$x(n)$triangular?$x(0)=1$, an empty product, is a triangular number. So are$x(2)=2\times 3=6$, and$x(4)=2\times 3\times 5\times 7=210$. 0answers 18 views ### Prove the following property for the given sequence. Consider the following sequence: https://oeis.org/A079523 Let$a(n)$denote the terms of the sequence of utterly odd numbers. Show that for each$n$there exists a$p$such that for each$k\in \mathbb{...
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Consider the following sequence defined here https://oeis.org/A079523 of utterly odd numbers: These are numbers such whose binary representation ends in an odd number of ones. If $n$ is an utterly ...
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### Geometry: polar equation of conics

In finding polar equations of conics we take focus as the origin but ellipse has 2 focci so which focus is taken as the origin? Or If if focus is not taken on origin then how to find the polar ...
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### Distributions of prime numbers

When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers: [To see these ...