Questions tagged [experimental-mathematics]

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280 votes
47 answers
109k 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 ...
152 votes
52 answers
23k views

Experimental mathematics leading to major advances

I would like to ask about examples where experimentation by computers has led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples of ...
114 votes
3 answers
5k 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 $$\...
Nikita Kalinin's user avatar
103 votes
17 answers
15k views

Theorems that are essentially impossible to guess by empirical observation

There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold ...
90 votes
11 answers
13k 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 ...
76 votes
13 answers
8k 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 ...
Joseph Van Name's user avatar
72 votes
13 answers
10k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
49 votes
1 answer
8k views

What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?

I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
Ryan Hendricks's user avatar
45 votes
4 answers
1k 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 ...
43 votes
7 answers
4k 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 ...
41 votes
8 answers
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 ...
37 votes
12 answers
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? ...
32 votes
1 answer
915 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 ...
Pedro Namtior's user avatar
30 votes
6 answers
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 ...
28 votes
1 answer
2k views

Conjectures inspired by AI

Today in Nature a paper described how AI guided mathematicians to make highly non-trivial conjectures, which they managed to prove, one in Knot Theory involving a new invariant, the other in ...
25 votes
1 answer
719 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}} = \...
Victor P's user avatar
  • 353
24 votes
3 answers
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 ...
Ryan Budney's user avatar
  • 42.9k
20 votes
4 answers
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 ...
Dave R's user avatar
  • 856
18 votes
8 answers
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 ...
18 votes
1 answer
699 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{...
Vladimir Reshetnikov's user avatar
18 votes
0 answers
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 \...
Gro-Tsen's user avatar
  • 29.9k
17 votes
6 answers
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 ...
17 votes
3 answers
707 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 ...
Timothy Chow's user avatar
  • 78.1k
17 votes
1 answer
2k views

Why are Goldbach laggards biased towards $2 \bmod 6$?

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 ...
Aaron Meyerowitz's user avatar
13 votes
1 answer
1k 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^{...
Turbo's user avatar
  • 13.7k
12 votes
0 answers
575 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 $\sum a_i p^i$...
Gro-Tsen's user avatar
  • 29.9k
10 votes
4 answers
2k 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 ...
9 votes
3 answers
1k 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 ...
Fedor Petrov's user avatar
9 votes
5 answers
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 ...
CHM's user avatar
  • 207
9 votes
2 answers
624 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 ...
Niles's user avatar
  • 599
8 votes
2 answers
807 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 ...
Roland Bacher's user avatar
7 votes
1 answer
488 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 ...
vzn's user avatar
  • 529
7 votes
2 answers
702 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, ...
graveolensa's user avatar
7 votes
1 answer
183 views

Reporting inconclusive experimental searches

In many areas of mathematics it is informative to conduct numerical experiments. But, it not uncommon that the searches do not lead to the examples or data one was hoping for. Since the numerical ...
Thomas Sauvaget's user avatar
7 votes
0 answers
427 views

Dynamics of a curious bijection of $\mathbb N$

The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows: Given an ...
Roland Bacher's user avatar
6 votes
1 answer
385 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, \...
joro's user avatar
  • 24.2k
6 votes
0 answers
474 views

Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below). Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...
Sebastien Palcoux's user avatar
5 votes
2 answers
774 views

Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)

I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819) $$L(n)=\sum_{k=1}^n \lambda(...
Vincent Granville's user avatar
5 votes
1 answer
1k 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 $...
spinkus's user avatar
  • 151
5 votes
1 answer
349 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 ...
user142929's user avatar
5 votes
1 answer
1k 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 ...
Mare's user avatar
  • 25.8k
5 votes
1 answer
603 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 ...
We Pretty's user avatar
5 votes
1 answer
473 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 ...
joro's user avatar
  • 24.2k
4 votes
4 answers
2k 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
spyros's user avatar
  • 51
4 votes
1 answer
285 views

Why should we expect this odd behavior of negative binomial distributions?

In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then $$ \Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
Michael Hardy's user avatar
3 votes
4 answers
798 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 ...
John Bentin's user avatar
  • 2,427
3 votes
1 answer
561 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 ...
René Gy's user avatar
  • 485
3 votes
1 answer
487 views

abc-conjecture and positive definite kernels, again?

One formulation of the abc-conjecture is: $$\forall a,b \in \mathbb{N}: \frac{a+b}{\gcd(a,b)}< \operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2 $$ Let us define: $$K(a,b) := \frac{2(...
mathoverflowUser's user avatar
3 votes
1 answer
204 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 $\...
joro's user avatar
  • 24.2k
3 votes
0 answers
278 views

Math videos featuring interesting data animations

I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...
Vincent Granville's user avatar