Questions tagged [experimental-mathematics]
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78 questions
282
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47
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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 ...
102
votes
17
answers
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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 ...
53
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1
answer
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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 ...
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3
answers
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Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
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votes
13
answers
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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 ...
92
votes
11
answers
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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 ...
156
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52
answers
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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 ...
4
votes
1
answer
378
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Experimental mathematics in Ramanujan's work
The field of experimental mathematics has led to the discovery of numerous remarkable identities and relations, often using computer algebra systems. Somos' work on finding algebraic identities ...
- 51
3
votes
1
answer
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Increasing sequences and Wieferich primes
We are trying to show that primes of the form $a(n)$ can't be
Wieferich primes.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, ...
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41
votes
8
answers
5k
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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 ...
4
votes
1
answer
294
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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
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
-4
votes
1
answer
605
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Multiplicative Persistence - Highest persistence found? [closed]
tried to ask on the math reddit but got deleted due to my account being new.
Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
- 21
3
votes
1
answer
535
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abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\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(...
- 3,144
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votes
0
answers
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Zeros of the semiprimes
Let $P$ be the prime zeta function
$$
P (s) = \sum_{p\, \in\text{ primes}} \frac 1 {p^s} = \frac {1} {2^s} + \frac {1} {3^s} + \frac 1 {5^s} + \frac 1 {7^s} + \frac 1 {11^s} + \cdots
$$
and define the ...
- 1,903
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votes
0
answers
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Using Ehrhart polynomials to count primes?
As indicated below, one could use the Ehrhart polynomials of the simplex in number theory.
Here are the questions without context first:
Questions:
The sum $$\sum_{k=0}^t (-1)^k ( \operatorname{...
- 3,144
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votes
0
answers
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Are there any known Khinchin reals for which the asymptotics of "average" of their coefficients seems experimentally known?
We can define a Khinchin Real and recall the definition of Khinchin's Constant
A real number $r$ is a Khinchin real if given the simple continued fraction expansion of $r$ as
$$ r = a_0 + \cfrac{1}{...
- 2,689
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votes
0
answers
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Does there exist an $L$-function for any subset of $\mathbb{N}$?
Consider the following prime sum:
\begin{aligned}
\sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}}
\end{aligned}
whose spikes appear at the Riemann $\zeta$ zeros as shown here.
Taking these detected spikes (...
- 1,903
76
votes
13
answers
9k
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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 ...
115
votes
3
answers
5k
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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
$$\...
- 5,063
28
votes
1
answer
2k
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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 ...
2
votes
0
answers
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Primes of the form $a+b^k$ for $k=(a \bmod 2),\ldots,n$?
Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities
such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.
A few examples are:
$2+4995825^k$ is prime for $k=0,\...
- 17.9k
2
votes
0
answers
85
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Curious sequences associated to continuous fractions
Given a strictly positive initial rational number $x_0$ in $\mathbb Q_>$
we define a sequence $x_0,x_1,\ldots$ recursively by
setting $x_{n+1}=x_n+1/S(x_n)$
for $S(x)=a_0+a_1+\ldots+a_k$
where $[...
- 17.9k
19
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 ...
45
votes
4
answers
1k
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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 ...
18
votes
8
answers
1k
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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 ...
2
votes
1
answer
404
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Mysteries of Wolfram's rule 18
[Unfortunately, I made some mistakes in my original question. I tacitly corrected them wherever I found them.]
Wolfram's rule 18 gives rise to fractal patterns, but when started with two black cells ...
- 9,720
37
votes
12
answers
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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?
...
1
vote
0
answers
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Astonishing affinity of Wolfram's rule 110 to the numbers 2 and 7
I investigated the evolution of a single black cell on 1-dimensional grids with periodic boundary conditions of variable sizes $N$ under Wolfram's rule 110 which is the only one for which Turing ...
- 9,720
1
vote
1
answer
139
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A special kind of pseudo-garden eden states in cellular automata
I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$.
It is clear that for each rule $R$ and ...
- 9,720
7
votes
1
answer
191
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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 ...
- 2,901
43
votes
7
answers
4k
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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 ...
5
votes
2
answers
874
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(...
- 3,098
18
votes
3
answers
785
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 ...
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0
votes
0
answers
100
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Possible shifts in finite elementary cellular automata
I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first ...
- 9,720
6
votes
0
answers
479
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} \...
0
votes
1
answer
492
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New experiments involving Ramanujan primes: Benford's law
I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
3
votes
0
answers
283
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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 ...
- 3,098
2
votes
1
answer
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What is wrong with the experimental evidence against the semi strong perfect graph theorem?
We got experimental evidence against the semi strong perfect graph theorem
and would like to learn what is wrong with it.
From Recognizing the P4-structure of bipartite graph
The P4-structure of a ...
- 25.4k
7
votes
0
answers
429
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 ...
- 17.9k
1
vote
1
answer
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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 ...
- 1,183
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votes
3
answers
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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 ...
- 109k
32
votes
1
answer
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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 ...
- 403
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vote
0
answers
375
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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 } ...
- 97
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votes
0
answers
306
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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 ...
- 25.4k
10
votes
5
answers
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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 ...
- 217
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votes
0
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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 ...
25
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3
answers
2k
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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 ...
- 44.4k
5
votes
1
answer
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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 ...
6
votes
1
answer
388
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$\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, \...
- 25.4k
31
votes
6
answers
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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 ...