Questions tagged [experimental-mathematics]
The experimental-mathematics tag has no usage guidance.
51
questions
67
votes
9answers
6k views
What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
3
votes
0answers
268 views
Why the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$?
According to both sagemath and Magma the curve $x^2+y^2+y+1=0$ has only one point over $\mathbb{F}_{3^7}$.
The projective closure has only one point too.
Q1 What hypothesis are missing to not violate ...
42
votes
8answers
5k views
Examples of creative experiments by mathematicians in modern days
I'm reading Random Circles on a Sphere and the authors did the following to empirically check their results:
To make a partial test of the accuracy of the above approximations an experiment was ...
2
votes
1answer
208 views
When does this limiting ratio give a real root $x$ to the equation of the form $\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0$?
By searching the Inverse Symbolic Calculator, we appear to be able to make the following conjecture about a real root to the equation:
$$\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0 \tag{1}$$
Let the ...
43
votes
4answers
897 views
How to write computer-assisted mathematics well?
Much has been said about writting good papers in mathematics. A short google query yields countless sources of advice. This skill also appears to be quite transferrable between basic branches of ...
3
votes
0answers
201 views
Some statistics related to the abc conjecture
We did some statistics about the 14 million good abc triples below 10^18
taken from Bart de Smith site.
This was examining just the top of the iceberg, since the
interesting triples grow very likely ...
17
votes
8answers
1k views
Conceptual insights and inspirations from experimental and computational mathematics [duplicate]
I am interested in whether experiments on computers can help identifying new ideas or concepts in Mathematics. I am not talking about confirming particular conjectures up to certain numbers (for ...
5
votes
1answer
314 views
Gadgets as primality tests
From the literature, showed below, I know two gadgets that provide a way to know if a positive integer (a positive quantity of units) is composite or a prime number. I would like to know if in the ...
0
votes
0answers
116 views
Similar problems for the Dedekind psi function than those that are in the literature for the Euler's totient function: reference request or proposals
I know the linked articles from Wikipedia for the Euler's totient function and the Dedekind psi function. I know that are in the literature famous, interesting and difficult problems involving the ...
1
vote
0answers
150 views
The Knuth-Stolarsky Conjecture in Addition Chains
I would like some general feedback on an experiment I have run in the field of addition chains.
An addition chain for target integer $n$ is defined as:
$$1=a_0<a_1<\cdots<a_r=n \text{ with }...
5
votes
1answer
323 views
Modified Pascal's triangle
I posted this question to Mathematics Stack Exchange but got no answers. I hope that this question is advanced enough for this forum:
In Pascal's triangle, each number is the sum of the two numbers ...
3
votes
0answers
182 views
Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
3
votes
1answer
176 views
$\psi(2,1/6),\psi(4,1/6)$ in terms of zeta and pi only and another closed form for zeta
Let $\psi(n,x)$ denote the polygamma function.
In this answer Lucia gave linear relations for $\psi(m,1/3),\psi(m,1/6),\zeta(m+1)$.
The computer managed to find closed form for $\psi(2,1/6)$ and $\...
6
votes
1answer
338 views
$\pi$ in terms of polygamma
The computer found this, but couldn't prove it.
Let $\psi(n,x)$ denote the polygamma function.
With precision 500 decimal digits we have:
$$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \...
2
votes
1answer
222 views
Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$
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 $\...
5
votes
1answer
798 views
Periods in the trivial extension algebra of the incidence algebra of the divisor lattice
Definition of $C_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the ...
13
votes
2answers
434 views
Automated search for bijective proofs
In enumerative combinatorics, a bijective proof that $|A_n| = |B_n|$ (where $A_n$ and $B_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection ...
2
votes
0answers
110 views
Conjectures that can be tested with large numbers of Hecke eigenvalues of GSp(4) automorphic forms
As part of my thesis work I have proved Ibukiyama's conjecture implies something about $\mathrm{SO}(5)$ forms associated to certain lattices lifting to $\mathrm{GSp}(4)$ (This was originally a ...
2
votes
2answers
358 views
Expansion of inverse logarithmic integral in terms of lambert w
Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:
$\operatorname{li}^{-...
72
votes
13answers
6k views
What computational problems would be good proof-of-work problems for cryptocurrency mining?
What computational mathematics problems that could be used as proof-of-work problems for cryptocurrencies? To make this question easier to answer, I want proof-of-work systems that work in ...
32
votes
1answer
843 views
Strange convergence of Euler's series for $\zeta(2)$
Using Maple to compare $\pi^2$ and the partial sums of $6\sum_{n=0}^{\infty}\frac{1}{n^2}$ I have noticed something that appears strange.
For instance, let $S_{k}=6\sum_{n=0}^{k}\frac{1}{n^2}$ be ...
14
votes
1answer
546 views
Several conjectured identities for polylogarithms
I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured
$$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname{...
8
votes
3answers
970 views
May $p^3$ divide $(a+b)^p-a^p-b^p$?
Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$?
The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not ...
108
votes
3answers
4k views
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
2
votes
0answers
196 views
On the cardinality of the set of right-truncatable primes
We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
9
votes
2answers
500 views
Periodicity in iterated powers of sin, cos, exp
Given a complex number $z$, consider the sequence
\begin{align*}
a_0 & = 1\\
a_1 & = (cos(1))^z\\
a_n & = (cos(a_{n-1}))^z
\end{align*}
This question is about trying to understand ...
12
votes
5answers
2k views
Which journals publish experimental results in pure maths?
All pure mathematicians know that the goal is to produce insight, rather than to simply obtain results. However, it might sometimes be of value to disseminate largely empirical work. In the same ...
2
votes
0answers
239 views
Experimentation with partial Euler products
Richard Mathar $[1]\& [2]$ shows that
\begin{align}
&\zeta_{2}(s)\equiv\prod_{\Omega(n)=2}^{}\left(\dfrac{1}{1 - n^{s}}\right)^{-1}=\exp \left(\sum _{k=1}^n \frac{P(k s)^2+P(2 k s)}{2 k}\right)...
2
votes
1answer
463 views
Trying to prove a congruence for Stirling numbers of the second kind
This a repost of a question I asked at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement.
Proposition: When $n$ and $m$ are ...
5
votes
1answer
433 views
Many integral points on quartic models of elliptic curve via differences of squares
Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...
255
votes
45answers
101k views
Examples of unexpected mathematical images
I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
11
votes
0answers
459 views
Power series defined by Witt vectors / Teichmüller representatives of p-adics
Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$...
5
votes
1answer
977 views
Lines in image; are they significant to prime numbers if so how?
Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where $...
13
votes
0answers
2k views
Distribution of digits of $pq$-adic idempotents (aka “automorphic numbers”)
Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p \...
25
votes
1answer
660 views
“Harmonacci” recurrence and identities for $\pi$
While playing with something totally irrelevant I stumbled upon the recurrence:
$$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$
It turns out that given $a_0 = 1, a_1 = 1$,
$$lim \frac{a_{2n}}{a_{2n-1}} = \...
13
votes
1answer
880 views
Connection between Infinite continued fractions, elliptic integrals and AGM
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
22
votes
3answers
2k views
Persistent homology of Gaussian fields in Euclidean space
If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
7
votes
1answer
450 views
Experimental mathematics: how are floating point equations discovered/converted to exact equations?
the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
40
votes
7answers
3k views
Can pure mathematics harness citizen science?
Having just finished Michael Nielsen's book "Reinventing Discovery", I find myself wondering if there are ways that pure mathematics research can engage the public in the way that GalaxyZoo or Foldit ...
9
votes
5answers
1k views
What is the theoretical interest of finding closed-form solutions of infinite series?
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve it ...
3
votes
4answers
1k views
buffon needle experiment [closed]
Hi, what are the "best" values for lenght of needle (l) and distance between paralles (d) for an accurate approximation of pi? Does it have to be l-d-1.0 or ld? Thanx
34
votes
12answers
3k views
Interesting conjectures “discovered” by computers and proved by humans?
There are notable examples of computers "proving" results discovered by mathematicians, what about the opposite:
Are there interesting conjectures "discovered" by computers and proved by humans?
...
20
votes
4answers
2k views
Does the set of happy numbers have a limiting density?
A positive integer $n$ is said to be happy if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.
For example, 7 is ...
30
votes
6answers
2k views
Useful tricks in experimental mathematics
There are a few computational tricks which are useful in experimental mathematics.
These tricks are mostly very elementary and often only given as exercices in books.
A typical example is the ...
12
votes
1answer
1k views
Why are Goldbach laggards biased towards $2 \mod 6$?
For even $n$,let $g(n)$ be the number of ways to write $n$ as a sum of two primes $n=p+q$ with $p \le q$. Define $a_k$ to be the largest $n$ with $g(n)=k$. I would bet money that no-one will disprove (...
3
votes
4answers
734 views
What patterns have been measured in the graph of the number of two-prime-sum representations of even numbers?
There are remarkable patterns of density in the graph http://upload.wikimedia.org/wikipedia/commons/7/7c/Goldbach-1000000.png plotting the number of representations of even numbers up to a million as ...
2
votes
1answer
266 views
A determinant involving only cyclotomic factors
Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients
$x^{\alpha(i+j)}$ ...
8
votes
2answers
774 views
Modular congruences related to sums of Catalan numbers
I am curious if somebody can be helpful concerning the following
experimental observation:
There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and
$\beta_0,\beta_1,\dots$, both with values ...
10
votes
4answers
1k views
When Have Numerology and Computational Experimentation Been Successful?
When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of ...
7
votes
2answers
654 views
Infinite product experimental mathematics question.
A while ago I threw the following at a numerical evaluator (in the present case I'm using wolfram alpha)
$\prod_{v=2}^{\infty} \sqrt[v(v-1)]{v} \approx 3.5174872559023696493997936\ldots$
Recently, ...
