Questions tagged [experimental-mathematics]

The utilization of advanced computing technology in mathematical research: new mathematical results discovered partly or entirely with the aid of computer-based tools.

Experimental Mathematics is an approach to Mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.

Experimental Mathematics makes use of numerical methods to calculate approximate values for integrals and infinite series. Arbitrary precision arithmetic is often used to establish these values to a high degree of precision – typically $100$ significant figures or more. Integer relation algorithms are then used to search for relations between these values and mathematical constants. Working with high precision values reduces the possibility of mistaking a mathematical coincidence for a true relation. A formal proof of a conjectured relation will then be sought – it is often easier to find a formal proof once the form of a conjectured relation is known.

For example,

  • Roger Frye used experimental Mathematics techniques to find the smallest counterexample to Euler's sum of powers conjecture.
  • The ZetaGrid project was set up to search for a counterexample to the Riemann hypothesis.
197 questions
12
votes
2 answers

Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Recall that the Thue–Morse sequence$^{[1]}$$\!^{[2]}$$\!^{[3]}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its bitwise complement (i.e. obtained by flipping…
12
votes
1 answer

An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$

Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$ $$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{2332\,\pi ^6}{2679 075}\,4^{-3…
12
votes
1 answer

$1$ as difference of composites with same number of prime factors and smallest examples

It is probably open can we for every $k \in \mathbb N$ find two composites $a_k$ and $b_k$ such that $a_k$ and $b_k$ have exactly $k$ prime factors and $a_k-b_k=1$. Smallest examples found so far are: for $k=1$ $$3^2-2^3=1$$ for $k=2$ $$3 \cdot 5 -…
11
votes
5 answers

Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$

Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed: $$x^6+x^4+x^2+1=(x^2+1)(x^4+1)$$ Let…
11
votes
1 answer

On $_2F_1(\tfrac13,\tfrac23;\tfrac56;\tfrac{27}{32}) = \tfrac85$ and $_2F_1(\tfrac14,\tfrac34;\tfrac78;\tfrac{48}{49}) = \tfrac{\sqrt7}3(1+\sqrt2)$

Consider the rather interesting and new evaluations for $_2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac{n}{n+1}};z\right)$, $$\begin{aligned} _2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac23};\tfrac{2^2\times3^3}{121}\right) &=…
10
votes
4 answers

The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?

(The results below extend this post.) Given the Clausen function $\operatorname{Cl}_n\left(z\right)$. And, $$\begin{aligned} \operatorname{Cl}_2\left(\frac\pi2\right) &= \text{Catalan's constant}\\ \operatorname{Cl}_2\left(\frac\pi3\right) &=…
10
votes
1 answer

Statistics for $N$ in sum of cubes $a^3+b^3+c^3 = N^3$?

Q: What is the percentage of $n$ up to a bound $N$ such that, $$a^3+b^3+c^3 = n^3\tag1$$ has a solution in positive integers? The sequence A023042 shows a large percentage. I have extended that to…
9
votes
2 answers

What experimental-mathematical problem would you try to solve if you had a supercomputer?

At the present moment, what open mathematical problem do you seriously think you could solve if you had a very powerful computer at your disposition? I mean something like the Four-Color Problem, i.e. that is simple but more tractable with a…
Immanuel Weihnachten
  • 1,778
  • 1
  • 14
  • 26
9
votes
1 answer

Powers of a simple matrix and Catalan numbers

Consider $m \times m$ anti-bidiagonal matrix $M$ defined as: $$M_{ij} = \begin{cases} -1, & i+j=m\\ \,\,\ 1, & i+j=m+1\\ \,\,\, 0, & \text{otherwise} \end{cases}$$ Let $S_n$ stand for the sum of all elements of the $n$-th power of the…
9
votes
1 answer

Can we identify the limit of this arithmetic/geometric mean like iteration?

Let $a_0 = 1$ and $b_0 = x \ge 1$. Let $$ a_{n+1} = (a_n+\sqrt{a_n b_n})/2, \qquad b_{n+1} = (b_n + \sqrt{a_{n+1} b_n})/2. $$ Numeric computation suggests that regardless of the choice of $x$, $a_n$ and $b_n$ always converge to the same value. Can…
faceclean
  • 6,585
  • 8
  • 34
  • 63
9
votes
1 answer

How are Trott constants found; are there mathematical results?

While reading Steven Finch's wonderful book Mathematical Constants, I encountered the Trott constant which was presented as the real number such that the digits of its decimal expansion are the digits of its simple continued fraction expansion. This…
8
votes
0 answers
8
votes
0 answers

Is there an identity between the Clausen function $\rm{Cl}_8\left(\frac\pi3\right)$ and $\sum_{n=1}^\infty \frac{1}{n^9\,\binom {2n}n}$?

Given the log sine integral, $$\rm{Ls}_m\Big(\frac{\pi}3\Big) = \int_0^{\pi/3}\Big(\ln\big(2\sin\tfrac{\theta}{2}\big)\Big)^{m-1}\,d\theta$$ we have in this post, $$\begin{aligned} \frac\pi{2!}\,\rm{Ls}_2\Big(\frac{\pi}3\Big)…
8
votes
1 answer

Semiperimeter of isosceles Heronian triangles.

A Heronian triangle is a triangle with integer sides and area, named after Heron's formula which states that the area of a triangle with sides $a$, $b$, and $c$ is $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$ where $s = (a + b + c)/2$ is the semiperimeter of…
Peter Kagey
  • 4,789
  • 9
  • 32
  • 80
1 2
3
13 14