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3
votes
1answer
147 views

$\psi(2,1/6),\psi(4,1/6)$ 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 $\...
6
votes
1answer
266 views

$\pi$ 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, \...
2
votes
1answer
206 views

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 $\...
5
votes
1answer
300 views

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 ...
12
votes
2answers
289 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 ...
2
votes
0answers
94 views

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
votes
0answers
125 views

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}^{-...
65
votes
13answers
4k views

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
votes
1answer
718 views

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 ...
14
votes
1answer
457 views

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{...
8
votes
3answers
598 views

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 ...
99
votes
3answers
3k views

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 $$\...
1
vote
0answers
172 views

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}...
9
votes
2answers
438 views

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 ...
10
votes
4answers
1k 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 ...
2
votes
0answers
219 views

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)...
2
votes
1answer
342 views

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 ...
5
votes
1answer
396 views

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 ...
217
votes
37answers
91k views

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 ...
11
votes
0answers
339 views

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 $\sum a_i p^i$...
5
votes
1answer
845 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 $...
11
votes
0answers
2k 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 \...
24
votes
1answer
590 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}} = \...
13
votes
1answer
754 views

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^{...
7
votes
1answer
410 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 ...
37
votes
7answers
3k views

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 ...
9
votes
4answers
805 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
4answers
1k 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
34
votes
12answers
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? ...
20
votes
4answers
2k 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 ...
27
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
716 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
564 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
249 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 $x^{\alpha(i+j)}$ ...
8
votes
2answers
759 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 ...
10
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
613 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, ...
116
votes
46answers
16k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples ...