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.
11
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0answers
176 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 ...
8
votes
0answers
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 $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$
...
1
vote
0answers
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 $x(4)=2\times 3\times 5\times 7=210$.
0
votes
0answers
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{...
0
votes
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
votes
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 ...
5
votes
0answers
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 ...
3
votes
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 $\...
0
votes
0answers
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:
$$S_1 = \{(x,y) : | x + |x|| + |y + |y|| < 2\}$$
$$S_2 = \{(x,y) : | x - |x|| + |y - |y|| < 2\}$...
0
votes
0answers
12 views
Optimization with inputs of ordinal constrains
A simplified version of my question: What method I shall explore to estimation a equation as below:
$$\text{score}=X_i+Y_j+Z_k+\varepsilon$$
$i,j,k$ are discrete variables which takes values in a ...
5
votes
2answers
107 views
For any positive integer $k$, are there always primes $p$ and $q$ such that $q-p=2^k$?
Experimental evidence suggests to me that there are always primes $p$ and $q$ such that $q-p=2^k$.
Some examples include: $5-3=2$, $11-7=4$, $19-11=8$, $29-13=16$, $43-11=32$, etc.
I am now sure how ...
1
vote
0answers
22 views
Determine convergence from experimental data set
I have ran a simulation to get a set of data points represented in the image below:
Image
The $x$ values range from 5 to 400 in units of 1 (discrete x axis).
It is clear from the grapical ...
6
votes
1answer
101 views
What about sequences $\{\sum_{k=1}^n (\operatorname{rad}(k))^p\}_{n\geq 1}$ containing an infinitude of prime numbers, where $p\geq 1$ is integer?
We denote the radical of the integer $n> 1$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p,$$
taking $\operatorname{rad}(1)=1$ that is this definition from Wikipedia.
In ...
2
votes
2answers
66 views
Why this behaviour of primes?
I calculated $$(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$$
We can see that there are ...
12
votes
1answer
342 views
$1$ as difference of composites with same number of prime factors and smallest examples
It is probably open can we for every $k \in \mathbb N$ find two composites $a_k$ and $b_k$ such that $a_k$ and $b_k$ have exactly $k$ prime factors and $a_k-b_k=1$.
Smallest examples found so far are:...
0
votes
0answers
32 views
Can you propose a conjectural $\text{Upper bound}(x)$ for the counting function of a sequence of primes arising from the Eratosthenes sieve?
I hope that next question is clear here, since my English is bad. In next paragraph I define a sequence of prime numbers as a sum of an even number of prime numbers minus $1$, where previous terms, ...
11
votes
1answer
171 views
Conjectures Disproven by the use of Computers?
Question: Is there a list of conjectures (famous or not so famous) that were shown to be false by employing the use computers ?
This is just curiosity more than anything. I was actually wondering if ...
1
vote
2answers
99 views
Making something a control parameter or a variable when analysing a dynamical system
I am writing down a draft trying to accurately characterize some nonlinear/noninvertible discrete dynamical systems (of a former question here) and due to my lack of knowledge I am having doubts here ...
0
votes
1answer
48 views
On the solutions of an equation involving the Euler's totient function that is solved by the primes of Rassias' conjecture
Here $\varphi(n)$ denotes the Euler's totient function. I've deduced a family of prime numbers $$(x=p_1,y=p_2)$$ that solve an equation involving the Euler's totient function. These primes are the ...
7
votes
1answer
199 views
Skewes' number, and the smallest $x$ such that $\pi(x) > \operatorname{li}(x) - \tfrac1n \operatorname{li}(x^{1/2})$?
We know that Skewes' number is a yet unknown huge integer $x$ such that,
$$\pi(x) > \operatorname{li}(x)\tag1$$
where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the ...
9
votes
1answer
374 views
Powers of a simple matrix and Catalan numbers
Consider $m \times m$ anti-bidiagonal matrix $M$ defined as:
$$M_{ij} = \begin{cases}
-1, & i+j=m\\
\,\,\ 1, & i+j=m+1\\
\,\,\, 0, & \text{otherwise}
\end{cases}$$
Let $S_n$ stand ...
0
votes
0answers
18 views
Method for measuring distance of a irregular curved profile
I am trying to measure a curved profile of a surface(2D) to determine the surface availability at different rate of testing. I have attached an image for a rough picture.enter image description here
...
4
votes
1answer
144 views
Can this be continued forever?
We have:
$$1=1^2$$ and $$25=5^2=3^2+4^2$$
and $$441=21^2=20^2+4^2+5^2$$
So, for $k=1,2,3$ we have a $k$-digit number that is a perfect square and a sum of $k$ different non-zero perfect squares.
...
2
votes
2answers
86 views
Is there an infinite number of numbers like $1600$?
My reputation is at this moment at $1600$.
I did some experimenting with $1600$ and obtained the following:
Evidently, it is a perfect square $1600=40^2$
Also, it is a hypothenuse of a Pythagorean ...
15
votes
2answers
763 views
How would you prove that there is only a finite number of these primes?
For the purpose of this question you can assume/consider number $1$ to be a prime number, but the final result should not depend on that, that is, that there is only a finite number of primes like the ...
2
votes
1answer
93 views
find unique numbers to write on the faces of two dice so the sum is always a prime.
The puzzle is to find the, smallest*, unique**, whole, non-negative, numbers to write on the faces of two dice such that the numbers you see on each dice after rolling sum equals a prime number.(...
1
vote
0answers
52 views
Getting a wrong value of $\pi$ [duplicate]
I read this question somewhere
Let a square of length 1 unit and a circle is inscribed in it such that it touches the sides of square.
Now cut small portion(outside the circle) of square from each ...
8
votes
1answer
226 views
What's the maths behind the movements of a two-legged air dancer (aka skydancer, tube man)? How can I simulate its behavior?
I am trying to understand the maths behind the movements of a two-legged air dancer, aka skydancer aka tube man. Well, I mean these cheery friends here:
The following discrete time algorithm is my ...
10
votes
1answer
407 views
Statistics for $N$ in sum of cubes $a^3+b^3+c^3 = N^3$?
Q: What is the percentage of $n$ up to a bound $N$ such that,
$$a^3+b^3+c^3 = n^3\tag1$$
has a solution in positive integers?
The sequence A023042 shows a large percentage. I have extended that ...
22
votes
1answer
594 views
Why is this family of dynamical systems able to produce spirals and clusters of points?
I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a ...
0
votes
2answers
34 views
On variations of Rowland's sequence using the radical of an integer $\prod_{p\mid n}p$
This afternoon I tried to create a Rowland's sequence using the radical of an integer in my formula. I don't know if it was in the literature, but I know that also there were variations on Rowland's ...
3
votes
1answer
185 views
Origin of rapidly converging sequence for $e^\pi$
The following sequence converging to Gelfond's constant ($e^\pi$) is apparently mentioned (and originates?) in the book "Mathematics by Experiment: Plausible Reasoning in the 21st Century.", which I ...
0
votes
1answer
45 views
Can someone check something for me in the computational sense?
Here it is presented the sum which appeared in a recent mathematical competition at a local university: $$\sum_{j,k,l\geq0} \frac{1}{3^l\left(3^{j+k}+3^{k+l}+3^{l+j}\right)}$$ and it is said that the ...
2
votes
2answers
69 views
Has the diophantine equation $z^2+zx=2x^2y$ infinitely many solutions over integers $x,y,z\geq 2$?
After I did a reasoning involving even perfect numbers (see it if you want below my question) I am interested to know next
Question. Has $$z^2+zx=2x^2y\tag{D}$$ infinitelty many solutions $(x,y,z)$ ...
2
votes
1answer
159 views
The $\operatorname{rad}(3x+1)$ problem, where $\operatorname{rad}(n)$ is the radical of an integer
Let for integers $n\geq 1$, the radical of an integer $\operatorname{rad}(n)$, see the definition from this Wikipedia. Is a famous multiplicative function since it is related to the abc conjecture. I ...
2
votes
2answers
158 views
A fun fact relating to Goldbach Conjecture
I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and ...
0
votes
1answer
97 views
Generalised of $\pi={\prod_{n=1}^{\infty}\left(1+{1\over 4n^2-1}\right)\over \sum_{n=1}^{\infty}{1\over 4n^2-1}}$ in term of Fibonacci number
Given the Wallis's product of $\pi$,
$${\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}\...
2
votes
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 ...
1
vote
0answers
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:
$$\begin{matrix} \Phi\colon&\mathbb ...
0
votes
1answer
168 views
Does this normalization of the Riemann zeta function make sense?
For $c=0$ the following should be true for the $n$-th Gram point:
$$\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{\...
9
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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 ...
