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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Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th nontrivial zero of the Riemann zeta function, and…
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Conjectures Disproven by the use of Computers?

Question: Is there a list of conjectures (famous or not so famous) that were shown to be false by employing the use of computers? This is just curiosity more than anything. I was actually wondering if more often than not - computers show many…
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How would you prove that there is only a finite number of these primes?

For the purpose of this question you can assume/consider number $1$ to be a prime number, but the final result should not depend on that, that is, that there is only a finite number of primes like the one I found. I was trying by pure guesswork and…
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Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

There are few known closed form for values of the dilogarithm at specific points. Sometimes only the real part or only the imaginary part of the value is known, or a relation between several different values is known: [1][2][3][4][5][6]. Discovering…
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Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from computer experiments for example) that the following holds?…
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Extracting an asymptotic from a sequence defined by a recurrence relation

Suppose I have a sequence defined via its first term and a recurrence relation involving summation over all previous values with some coefficients. Here is the sequence I am interested in right now (although I would like to find a general method…
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On binomial sums $\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$ and log sine integrals

Seven years ago, I asked about closed-forms for the binomial sum $$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$ Some alternative results have been made. Up to a certain $k$, it seems it can be expressed surprisingly by a log sine…
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The number $\pi$ in an unexpected context

[This is a follow-up question to this one: Figures and Numbers: Relating properties of geometric shapes and their Fourier series.] Drawing shapes by some predefined Fourier series I found this square with rounded corners which is given by the…
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Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ with $a(t)$, $b(t)$ being $2\pi$-periodic…
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Yet another nested radical

Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$ I believe I can prove (with some handwaving) that $F$ does converge everywhere in $\mathbb{C}$ $\Im F = 0$ for sufficiently large real $x$ (actually larger than $x0 \approx…
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On Reshetnikov's integral $\int_0^1\frac{dx}{\sqrt[3]x\ \sqrt[6]{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\,|\alpha|}$

V. Reshetnikov gave the remarkable integral, $$\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})\tag1$$ More generally, given some integer/rational $N$, we are…
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Yet another conjecture about primes

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Conjecture: $\mathcal{N}(n!)-n!\:$ is either $1$ or a prime. It holds for n=1 to 99 and the expression is 1 for 3,11,27,37,41,73,77 and primes for all other $n<100$. I have no idea how to…
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Websites that promote co-operation and social networking among mathematicians

Are there some websites that could be defined as social networks for mathematicians and scientists? What I have in mind is something similar to Academia.edu or ResearchGate, but more specific towards mathematics and less formal ("Facebook-type", so…
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Prove that $\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$

Prove $$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$ Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise $$\int_0^\infty\Big(\arctan…
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Relationship between primes and practical numbers

This is my first post here. I am a musician, and not a mathematician, but I enjoy doing things to prime numbers and seeing what comes out. I have defined a sequence which takes the following values for $n$: -1 if $n$ is prime 1 if $n$ is a…
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