# Questions tagged [experimental-mathematics]

The experimental-mathematics tag has no usage guidance.

78
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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 ...

3
votes

1
answer

149
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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, ...

2
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0
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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}{...

3
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1
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347
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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 ...

0
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0
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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{...

50
votes

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 ...

2
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0
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194
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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 ...

0
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0
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196
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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 (...

4
votes

1
answer

287
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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{ ...

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votes

1
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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 ...

3
votes

1
answer

511
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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(...

0
votes

1
answer

121
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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 ...

2
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0
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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,\...

2
votes

1
answer

392
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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 ...

0
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0
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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 ...

2
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0
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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 $[...

1
vote

0
answers

370
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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 ...

7
votes

1
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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 ...

3
votes

0
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280
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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 ...

6
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0
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### 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} \...

102
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17
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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 ...

2
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1
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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 ...

28
votes

1
answer

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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 ...

0
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1
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474
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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 ...

7
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0
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### 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 ...

72
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 ...

5
votes

2
answers

806
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### 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(...

91
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11
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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 ...

3
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0
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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 ...

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8
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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 ...

1
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1
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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 ...

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4
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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 ...

3
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0
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### 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 ...

18
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8
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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 ...

5
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1
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349
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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 ...

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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 ...

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0
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367
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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 } ...

5
votes

1
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620
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### 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
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0
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### 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
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1
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### $\psi(2,1/6),\psi(4,1/6)$ 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 $\...

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1
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387
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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, \...

1
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1
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### 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 $\...

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1
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### 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 ...

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3
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### 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
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### 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
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2
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### 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}^{-...

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13
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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 ...

32
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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 ...

18
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1
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### 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{...

9
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3
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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 ...