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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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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 ...
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180 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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
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87 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 ...
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1answer
58 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 ...
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174 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 ...
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49 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 ...
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69 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 $x(4)=2\times 3\times 5\times 7=210$.
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17 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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1answer
46 views

About Utterly odd numbers.

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 ...
2
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1answer
30 views

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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113 views

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 ...
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1answer
102 views

Distribution of triangular, square, and pentagonal numbers

I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...
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40 views

Lines and Points

Let $S_1$ and $S_2$ be the set of points in x-y plane (where $x>0$ and $y$ is real) defined as: $$S_1 = \{(x,y) : | x + |x|| + |y + |y|| < 2\}$$ $$S_2 = \{(x,y) : | x - |x|| + |y - |y|| < 2\}$...
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12 views

Optimization with inputs of ordinal constrains

A simplified version of my question: What method I shall explore to estimation a equation as below: $$\text{score}=X_i+Y_j+Z_k+\varepsilon$$ $i,j,k$ are discrete variables which takes values in a ...
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2answers
107 views

For any positive integer $k$, are there always primes $p$ and $q$ such that $q-p=2^k$?

Experimental evidence suggests to me that there are always primes $p$ and $q$ such that $q-p=2^k$. Some examples include: $5-3=2$, $11-7=4$, $19-11=8$, $29-13=16$, $43-11=32$, etc. I am now sure how ...
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22 views

Determine convergence from experimental data set

I have ran a simulation to get a set of data points represented in the image below: Image The $x$ values range from 5 to 400 in units of 1 (discrete x axis). It is clear from the grapical ...
6
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1answer
101 views

What about sequences $\{\sum_{k=1}^n (\operatorname{rad}(k))^p\}_{n\geq 1}$ containing an infinitude of prime numbers, where $p\geq 1$ is integer?

We denote the radical of the integer $n> 1$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p,$$ taking $\operatorname{rad}(1)=1$ that is this definition from Wikipedia. In ...
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2answers
66 views

Why this behaviour of primes?

I calculated $$(p_k \cdot p_{k+1})\mod p_{k+2}$$ for $k=1,2,...25$ and obtained a following numbers $$1,1,2,12,7,12,1,2,16,11,40,12,24,7,13,16,48,40,72,48,40,60,15,48,12$$ We can see that there are ...
12
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1answer
342 views

$1$ as difference of composites with same number of prime factors and smallest examples

It is probably open can we for every $k \in \mathbb N$ find two composites $a_k$ and $b_k$ such that $a_k$ and $b_k$ have exactly $k$ prime factors and $a_k-b_k=1$. Smallest examples found so far are:...
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32 views

Can you propose a conjectural $\text{Upper bound}(x)$ for the counting function of a sequence of primes arising from the Eratosthenes sieve?

I hope that next question is clear here, since my English is bad. In next paragraph I define a sequence of prime numbers as a sum of an even number of prime numbers minus $1$, where previous terms, ...
11
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1answer
171 views

Conjectures Disproven by the use of Computers?

Question: Is there a list of conjectures (famous or not so famous) that were shown to be false by employing the use computers ? This is just curiosity more than anything. I was actually wondering if ...
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2answers
99 views

Making something a control parameter or a variable when analysing a dynamical system

I am writing down a draft trying to accurately characterize some nonlinear/noninvertible discrete dynamical systems (of a former question here) and due to my lack of knowledge I am having doubts here ...
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1answer
48 views

On the solutions of an equation involving the Euler's totient function that is solved by the primes of Rassias' conjecture

Here $\varphi(n)$ denotes the Euler's totient function. I've deduced a family of prime numbers $$(x=p_1,y=p_2)$$ that solve an equation involving the Euler's totient function. These primes are the ...
7
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1answer
199 views

Skewes' number, and the smallest $x$ such that $\pi(x) > \operatorname{li}(x) - \tfrac1n \operatorname{li}(x^{1/2})$?

We know that Skewes' number is a yet unknown huge integer $x$ such that, $$\pi(x) > \operatorname{li}(x)\tag1$$ where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the ...
9
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1answer
374 views

Powers of a simple matrix and Catalan numbers

Consider $m \times m$ anti-bidiagonal matrix $M$ defined as: $$M_{ij} = \begin{cases} -1, & i+j=m\\ \,\,\ 1, & i+j=m+1\\ \,\,\, 0, & \text{otherwise} \end{cases}$$ Let $S_n$ stand ...
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0answers
18 views

Method for measuring distance of a irregular curved profile

I am trying to measure a curved profile of a surface(2D) to determine the surface availability at different rate of testing. I have attached an image for a rough picture.enter image description here ...
4
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1answer
144 views

Can this be continued forever?

We have: $$1=1^2$$ and $$25=5^2=3^2+4^2$$ and $$441=21^2=20^2+4^2+5^2$$ So, for $k=1,2,3$ we have a $k$-digit number that is a perfect square and a sum of $k$ different non-zero perfect squares. ...
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2answers
86 views

Is there an infinite number of numbers like $1600$?

My reputation is at this moment at $1600$. I did some experimenting with $1600$ and obtained the following: Evidently, it is a perfect square $1600=40^2$ Also, it is a hypothenuse of a Pythagorean ...
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2answers
763 views

How would you prove that there is only a finite number of these primes?

For the purpose of this question you can assume/consider number $1$ to be a prime number, but the final result should not depend on that, that is, that there is only a finite number of primes like the ...
2
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1answer
93 views

find unique numbers to write on the faces of two dice so the sum is always a prime.

The puzzle is to find the, smallest*, unique**, whole, non-negative, numbers to write on the faces of two dice such that the numbers you see on each dice after rolling sum equals a prime number.(...
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0answers
52 views

Getting a wrong value of $\pi$ [duplicate]

I read this question somewhere Let a square of length 1 unit and a circle is inscribed in it such that it touches the sides of square. Now cut small portion(outside the circle) of square from each ...
8
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1answer
226 views

What's the maths behind the movements of a two-legged air dancer (aka skydancer, tube man)? How can I simulate its behavior?

I am trying to understand the maths behind the movements of a two-legged air dancer, aka skydancer aka tube man. Well, I mean these cheery friends here: The following discrete time algorithm is my ...
10
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1answer
407 views

Statistics for $N$ in sum of cubes $a^3+b^3+c^3 = N^3$?

Q: What is the percentage of $n$ up to a bound $N$ such that, $$a^3+b^3+c^3 = n^3\tag1$$ has a solution in positive integers? The sequence A023042 shows a large percentage. I have extended that ...
22
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1answer
594 views

Why is this family of dynamical systems able to produce spirals and clusters of points?

I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a ...
0
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2answers
34 views

On variations of Rowland's sequence using the radical of an integer $\prod_{p\mid n}p$

This afternoon I tried to create a Rowland's sequence using the radical of an integer in my formula. I don't know if it was in the literature, but I know that also there were variations on Rowland's ...
3
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1answer
185 views

Origin of rapidly converging sequence for $e^\pi$

The following sequence converging to Gelfond's constant ($e^\pi$) is apparently mentioned (and originates?) in the book "Mathematics by Experiment: Plausible Reasoning in the 21st Century.", which I ...
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1answer
45 views

Can someone check something for me in the computational sense?

Here it is presented the sum which appeared in a recent mathematical competition at a local university: $$\sum_{j,k,l\geq0} \frac{1}{3^l\left(3^{j+k}+3^{k+l}+3^{l+j}\right)}$$ and it is said that the ...
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2answers
69 views

Has the diophantine equation $z^2+zx=2x^2y$ infinitely many solutions over integers $x,y,z\geq 2$?

After I did a reasoning involving even perfect numbers (see it if you want below my question) I am interested to know next Question. Has $$z^2+zx=2x^2y\tag{D}$$ infinitelty many solutions $(x,y,z)$ ...
2
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1answer
159 views

The $\operatorname{rad}(3x+1)$ problem, where $\operatorname{rad}(n)$ is the radical of an integer

Let for integers $n\geq 1$, the radical of an integer $\operatorname{rad}(n)$, see the definition from this Wikipedia. Is a famous multiplicative function since it is related to the abc conjecture. I ...
2
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2answers
158 views

A fun fact relating to Goldbach Conjecture

I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and ...
0
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1answer
97 views

Generalised of $\pi={\prod_{n=1}^{\infty}\left(1+{1\over 4n^2-1}\right)\over \sum_{n=1}^{\infty}{1\over 4n^2-1}}$ in term of Fibonacci number

Given the Wallis's product of $\pi$, $${\pi\over 2}=\prod_{n=1}^{\infty}{4n^2\over 4n^2-1}=\prod_{n=1}^{\infty}\left(1+{1\over 4n^2-1}\right)\tag1$$ $${1\over 2}=\sum_{n=1}^{\infty}{1\over 4n^2-1}\...
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0answers
30 views

Sequences of the form $\frac{(a(n))^{b(n)}-1}{a(n)-1}$ as generators of primes: specify when such sequence is rich in primes and provide us an example

For integers $n\geq 1$, let $a:=\left\{ a(n) \right\}_{n=1}^\infty $ and $b:= \left\{ b(n) \right\}_{n=1}^\infty $ fixed sequences of integers with general terms $a(n)\geq 2$ and $b(n)\geq 2$. And ...
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65 views

About the study of an arithmetic function with peculiar behaviour

Preliminaries. Apart from this first definition, what follows can be skipped for the reader in a hurry. Definition 1. Let's define a function $\Phi$ as such: $$\begin{matrix} \Phi\colon&\mathbb ...
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1answer
168 views

Does this normalization of the Riemann zeta function make sense?

For $c=0$ the following should be true for the $n$-th Gram point: $$\frac{x}{2 \pi e}\log\left(\frac{x}{2 \pi e}\right) = \frac{x}{2 \pi e}\log \left(\frac{x}{2 \pi e}\right) + \frac{-c+n}{e}-\frac{\...
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1answer
92 views

Can we identify the limit of this arithmetic/geometric mean like iteration?

Let $a_0 = 1$ and $b_0 = x \ge 1$. Let $$ a_{n+1} = (a_n+\sqrt{a_n b_n})/2, \qquad b_{n+1} = (b_n + \sqrt{a_{n+1} b_n})/2. $$ Numeric computation suggests that regardless of the choice of $x$, $a_n$ ...
16
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0answers
361 views

Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar ...
4
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1answer
66 views

Set conjectures concerning the asymptotic behaviour of erratic arithmetic functions, related to the Möbius function and the Liouville function

I get from an artificious way, but simple, two similar statements that I belive that are true (I had that identify some arithmetic functions), that is my Claim below. In next question I am asking ...
3
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1answer
133 views

How can we show that $\int_{-1}^{1}{1\over x}\sqrt{1+x\over 1-x}\ln\left({1-x+2x^3\over 1+x-2x^3}\right)dx=2\pi arccot\left(2\sqrt{\phi^3}\right)?$

Motivated by this question $$\int_{-1}^{1}{1\over x}\sqrt{1+x\over 1-x}\ln\left({1-x+2x^3\over 1+x-2x^3}\right)\mathrm dx=\color{blue}{2\pi \operatorname*{arccot}\left(2\sqrt{\phi^3}\right)}\tag1$$ ...
4
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1answer
96 views

How can we show that $\sum_{k=1}^{n}{2n+2\choose 2k}B_{2k}=n?$

Given the sum $$\sum_{k=1}^{n}{2n+2\choose 2k}B_{2k}=n\tag1$$ Where $B_{2k}$ is Bernoulli number It is quite interesting to me, the answer results in a natural number, how do you go about ...
1
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1answer
42 views

Finding the day of the same date of previous month?

I wanted to know if there was a short way that I could determine the day which would have been for a given date. For example, looking at the date today 4/3/17. Is there anyway we can easily ...