The experimental-mathematics tag has no wiki summary.
4
votes
1answer
386 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 ...
6
votes
0answers
81 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 ...
22
votes
1answer
364 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 $a_0 = 1, a_1 = 1$,
$$lim \frac{a_{2n}}{a_{2n-1}} = ...
7
votes
1answer
286 views
Experimental mathematics: how are floating point equations discovered/converted to exact equations?
the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
8
votes
4answers
630 views
What is the theoretical interest of finding closed-form sols. of infinite series?
Hi,
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve ...
3
votes
5answers
934 views
buffon needle experiment [closed]
Hi, what are the "best" values for lenght of needle (l) and distance between paralles (d) for an accurate approximation of pi? Does it have to be l-d-1.0 or ld? Thanx
23
votes
11answers
2k views
Interesting conjectures “discovered” by computers and proved by humans?
There are notable examples of computers "proving" results discovered by mathematicians, what about the opposite:
Are there interesting conjectures "discovered" by computers and proved by humans?
...
18
votes
4answers
1k views
Does the set of happy numbers have a limiting density?
A positive integer $n$ is said to be happy if the sequence
$$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 ...
23
votes
6answers
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 ...
11
votes
0answers
481 views
Why are Goldbach laggards biased towards $2 \mod 6$?
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 ...
2
votes
2answers
444 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 ...
2
votes
1answer
215 views
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
$x^{\alpha(i+j)}$ ...
7
votes
2answers
627 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 ...
8
votes
4answers
1k 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 ...
6
votes
2answers
525 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, ...
49
votes
33answers
8k views
Experimental Mathematics
I would like to ask about examples where experimentation by computers have led to major mathematical advances.
Motivation
I am aware about a few such cases and I think it will be useful to gather ...
