Mathematics Stack Exchange
Mathematics Stack Exchange

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.
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If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, would it still be considered unfair for $50$…
Adam Freymiller
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Conjectured formula for the Fabius function

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a poet or philosopher could say "it knows and…
Vladimir Reshetnikov
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Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$…
Simon Parker
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Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, Comparing $2013!$ and $1007^{2013}$ or yet the…
Start wearing purple
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Mind-blowing mathematics experiments

We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very counter-intuitive but can be explained using mathematics? I…
Namnyef
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Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

I encountered the following integral and numerical approximations tentatively suggest that it might have a simple closed…
Vladimir Reshetnikov
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Free online mathematical software

What are the best free user-friendly alternatives to Mathematica and Maple available online? I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, even though the calculation time there was…
Start wearing purple
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On the integral $\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

V. Reshetnikov gave the interesting integral, $$\int_0^1\frac{\mathrm dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{2-x\,\sqrt3}}=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ After some experimentation, it turns out that more generally, given some…
Tito Piezas III
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Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of Malmsten—Vardi—Adamchik integrals). It appears…
Vladimir Reshetnikov
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A number $n$ which is the sum of all numbers $k$ with $\sigma(k)=n$?

For a positive integer $n$, let us define a set $$A_n = \{ k\in\mathbb{N} \mid \sigma(k) = n \}$$ where $\sigma$ is the divisor-sum function (a well-known multiplicative number-theoretic function). Clearly $A_n \subseteq \{ 1,2,3,\ldots,n\}$ (since…
Jeppe Stig Nielsen
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Why is this family of dynamical systems able to produce spirals and clusters of points?

I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a complex discrete map as follows: (Disclaimer, tl:dr:…
iadvd
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Most wanted reproducible results in computational algebra

I am interested in suggestions for major computational results obtained with the help of mathematical software but not easily verifiable using computers. "Most wanted" could refer, for example, to the following: results which are highly…
Olexandr Konovalov
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Is there a real number lookup algorithm or service?

Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of something like Sloane's Online Encyclopedia of…
Simon Nickerson
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Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$

I was experimenting with hypergeometric-like series and discovered the following conjecture (so far confirmed by more than $5000$ decimal…
Vladimir Reshetnikov
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Convergence acceleration technique for $\zeta(4)$ (or $\eta(4)$) via creative telescoping?

Question Is it already known whether the $\zeta(4):=\sum_{n=1}^{\infty}1/n^4$ accelerated convergence series $(1)$, proved for instance in [1, Corollaire 5.3], could be obtained by a similar technique to the ones explained by Alf van der Poorten…
Américo Tavares
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