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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On the solutions of an equation involving the Euler's totient function that is solved by the primes of Rassias' conjecture

Here $\varphi(n)$ denotes the Euler's totient function. I've deduced a family of prime numbers $$(x=p_1,y=p_2)$$ that solve an equation involving the Euler's totient function. These primes are the prime numbers that appear in the Rassias'…
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Method for measuring distance of a irregular curved profile

I am trying to measure a curved profile of a surface(2D) to determine the surface availability at different rate of testing. I have attached an image for a rough picture.enter image description here Actually I have a reference geometry and performed…
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On variations of Rowland's sequence using the radical of an integer $\prod_{p\mid n}p$

This afternoon I tried to create a Rowland's sequence using the radical of an integer in my formula. I don't know if it was in the literature, but I know that also there were variations on Rowland's recursion in the literature. Definition. I define…
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Can someone check something for me in the computational sense?

Here it is presented the sum which appeared in a recent mathematical competition at a local university: $$\sum_{j,k,l\geq0} \frac{1}{3^l\left(3^{j+k}+3^{k+l}+3^{l+j}\right)}$$ and it is said that the answer is $9/8$. There also the answer is given…
user480281
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Generalised of $\pi={\prod_{n=1}^{\infty}\left(1+{1\over 4n^2-1}\right)\over \sum_{n=1}^{\infty}{1\over 4n^2-1}}$ in term of Fibonacci number

Given the Wallis's product of $\pi$, $${\pi\over 2}=\prod_{n=1}^{\infty}{4n^2\over 4n^2-1}=\prod_{n=1}^{\infty}\left(1+{1\over 4n^2-1}\right)\tag1$$ $${1\over 2}=\sum_{n=1}^{\infty}{1\over 4n^2-1}\tag2$$ $(2)$ is a telescope sum. We can rewrite…
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Does this normalization of the Riemann zeta function make sense?

For $c=0$ the following should be true for the $n$-th Gram point: $$\frac{x}{2 \pi e}\log\left(\frac{x}{2 \pi e}\right) = \frac{x}{2 \pi e}\log \left(\frac{x}{2 \pi e}\right) + \frac{-c+n}{e}-\frac{\vartheta (x)}{\pi e}$$ and for $c=\frac{1}{2}$ it…
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Describing the path of a particle on a wheel that changes direction

So I start with a particle rolling along the outside of a wheel. This can be constructed by setting up a vector function for a circle and adding a constant velocity to one component. My wheel is going to be moving horizontally with a velocity v_0,…
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Is this plot linear in the same way the Chebyshev function is linear?

The Dirichlet generating function for the von Mangoldt function is: $$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right) \;\;\;\;\;\;\;\;\;(1)$$ where: $$\frac{\zeta (c) \zeta (s)}{\zeta…
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Can we prove that plumb line is vertical to ground?

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument. I was wondering how could one assume at that time that the…
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A computational experiment about identities involving the sum of remainders function

Let $\sigma(m)$ the sum of divisors function and $$S(m)=\sum_{k=1}^m\text{m mod k}$$ the sum of remainders function, then it is know that for integers $m>1$ $$\sigma(m)+S(m)=S(m-1)+2m-1.$$ On the other hand since the sum of divisor functioin is…
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Method to study obvious properties

Most of the time studying mathematics we come across various properties like associative, commutative,...etc. These properties are obvious and sometimes I feel why at all they are given in the text. One such eg. is ! one such example of the…
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How to calculate these particular values in this experiment?

I am running a statistics experiment where I need to calculate certain values. However, this is where I am having difficulty. Here is how the experiment works: Two Individuals Participate There is a spinner with 6 labeled tiles[1-6] and a regular…
Veer Singh
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Balanced Latin Square

For making a good Between-Object user study, this is suggested to use a Latin Square to give all the different conditions, different order of representation of those conditions. However, when the number of conditions grow, the number of…
lonesome
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Can a simple plot be used as a proof-without-words?

Can this simple plot be used as a proof-without-words? Edit "No, it suggests but does not prove." Plot of $2^{1 + n} = 1 + 3^n:$ Motivated by this question, I reworked the non-loopback inequalities for Collatz and Waring into an equality: $2^{1 +…
Fred Kline
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What do these contour maps tell me about my Collatz expression?

I tested this limit on WolframAlpha, $$ \lim_{t\to\infty}\frac {2\ 3^r (2 t - 1) - 6} {3\ 2^r (2 t - 1) - 6}=\left(\frac{3}{2}\right)^{r-1},$$ which displayed two contour maps: . Can someone describe the what we are seeing? Anything notable?…
Fred Kline
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