Questions tagged [experimental-mathematics]
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78 questions
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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 ...
- 25.4k
18
votes
1
answer
727
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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{...
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5
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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 ...
- 51
10
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4
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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 ...
5
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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 ...
- 26.5k
2
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2
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655
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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:
$\operatorname{li}^{-...
- 1,903
3
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0
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Growth functions of finite group - computation, typical behaviour, surveys?
Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour:
Rubik's growth in LOG scale (see MO322877):
S_n n=9 growth and nice fit by normal ...
- 24.9k
3
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1
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$\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 $\...
- 25.4k
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votes
4
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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 ...
- 856
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2
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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 ...
- 609
17
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1
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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 ...
- 30.1k
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 $\...
- 9,730
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1
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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 ...
- 529
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4
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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 ...
- 2,437
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1
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"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 $a_0 = 1, a_1 = 1$,
$$lim \frac{a_{2n}}{a_{2n-1}} = \...
- 353
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1
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Lines in image; are they significant to prime numbers if so how?
Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where $...
- 151
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Distribution of digits of $pq$-adic idempotents (aka "automorphic numbers")
Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p \...
- 32.5k
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Connection between Infinite continued fractions, elliptic integrals and AGM
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
- 13.9k
2
votes
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 ...
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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
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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 ...
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2
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810
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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 ...
- 17.9k
2
votes
0
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237
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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}...
- 8,842
5
votes
1
answer
484
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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 ...
- 25.4k
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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 $\sum a_i p^i$...
- 32.5k
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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)...
- 1,903
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2
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714
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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, ...
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1
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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
$x^{\alpha(i+j)}$ ...
- 17.9k