Questions tagged [experimental-mathematics]
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81 questions
12
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Why does the Fourier transform of $μ(n)/n$ look like this?
Out of curiosity (and in relation to this MSE question), I computed numerically (an approximation to) the Fourier transform of $\mu(n)/n$ where $\mu$ is the Möbius function, viꝫ. $f\colon t \mapsto \...
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54
votes
18
answers
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Examples for the use of AI and especially LLMs in notable mathematical developments
The purpose of this question is to collect examples where large language models (LLMs) like ChatGPT have led to notable mathematical developments.
The emphasis in this question is on LLMs, but ...
15
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6
answers
2k
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Computationally inspired theorems and conjectures in knot theory
I am collecting examples of theorems and conjectures in knot theory that were originally discovered (or inspired) by computer experiments.
Examples include:
Hoste’s conjecture on zeros of the ...
- 476
52
votes
2
answers
2k
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Gap in binomial coefficients
Let $a,b$ be positive integers. Because binomial coefficients are integers, we know that $a!b!$ divides $(a+b)!$. For particular $a$ and $b$ there may be a gap $g$ with a tighter result, so $a!b!$ ...
- 4,609
15
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3
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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 ...
- 3,189
3
votes
1
answer
228
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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, ...
- 25.8k
3
votes
1
answer
158
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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
$$ r = a_0 + \cfrac{1}{...
- 3,189
4
votes
1
answer
506
views
Experimental mathematics in Ramanujan's work
The field of experimental mathematics has led to the discovery of numerous remarkable identities and relations, often using computer algebra systems. Somos' work on finding algebraic identities ...
- 51
67
votes
2
answers
12k
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What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
- 951
2
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0
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Zeros of the semiprimes
Let $P$ be the prime zeta function
$$
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
$$
and define the ...
- 1,913
0
votes
0
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251
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Does there exist an $L$-function for any subset of $\mathbb{N}$?
Consider the following prime sum:
\begin{aligned}
\sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}}
\end{aligned}
whose spikes appear at the Riemann $\zeta$ zeros as shown here.
Taking these detected spikes (...
- 1,913
4
votes
1
answer
356
views
Why should we expect this odd behavior of negative binomial distributions?
In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
-4
votes
1
answer
1k
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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
623
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(...
1
vote
1
answer
218
views
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 ...
- 9,982