Articles of experimental mathematics

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: An equilateral triangle can be divided into 3 congruent shapes. Questions: 1) a triangle can be divided into 3 congruent shapes. Is it equilateral? 2) a shape in the plane can be divided into n congruent shapes for any positive integer n. What can it […]

Divide by a number without dividing.

Can anyone come up with a way to divide any given x by any given y without actually dividing? For example to add any given x to any given y without adding you would just do: $x-(-y)$ And to subtract any given x from any given y (that is, y-x) you could do: $y+xe^{iπ}$ *edit: […]

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at where I posted a table of “convergents constants” (defined at for a few numbers. It would be nice to support the article with some quality analysis. Before June 9, 2011, was starting to extract and clearly define a pattern to these constants cf the article. […]

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 $_2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac{n}{n+1}};z\right)$, $$\begin{aligned} _2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac23};\tfrac{2^2\times3^3}{121}\right) &= \large\tfrac{\sqrt{33}}{3}\\[2mm] _2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac56};-\tfrac{135}{121}\right) &=\large\tfrac{\sqrt{33}}{10^{5/6}}\\[2mm] _2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac78};\tfrac{48}{49}\right) &= \tfrac{\sqrt7}3(1+\sqrt2)\\[2mm] _2F_1\left(\tfrac14,\tfrac34;\color{blue}{\tfrac9{10}};\tfrac{4}{5}\right) &=\large \tfrac1{5^{1/4}}\,\phi^{3/2}\end{aligned}$$ and golden ratio $\phi$, with the last a transformed version of Nemo’s answer. The transformation, $$_2F_1\left(\tfrac14,\tfrac34;c;\tfrac{4z(z-1)}{(1-2z)^2}\right)=\sqrt{1-2z}\,(1-z)^{1-c}\,_2F_1\left(-c+\tfrac32,\,c-\tfrac12;c;z\right)$$ allows it to be transformed to another form also with $a+b=-c+\tfrac32+c-\tfrac12 =1$. For example, the second one yields, […]

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$?

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and $\zeta(\frac{141}{23}) \approx e^\frac{1}{64}$. I also figured out that $\zeta(x)$ approaches $e^{2^{-x}}$ but I’m not sure that helps explain why these almost-equalities exist. How to quantify how surprising these almost-equalities are, and what is […]

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 a new identity of this sort is always of a great interest. I numerically discovered […]

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 limited to $60$ seconds. Very basic computations can be carried out with Wolfram Alpha. What if one is […]

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 classical $\pi^e$ vs $e^{\pi}$. In general I don’t like this kind of problems since a determined person with calculator can always find two numbers accidentally close to each other […]

Useful techniques of experimental mathematics (reference request)

I am searching for papers or books that explain thoroughly useful interesting techniques of experimental mathematics that can be understood and profitably applied by an undergraduate student.

On the integral $\int_0^1\frac{dx}{\sqrtx\ \sqrt{1-x}\ \sqrt{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

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 integer/rational $N$, we are to find an algebraic number $\gamma$ that solves, $$\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}\tag2$$ (Compare to the similar integral in this post.) Equivalently, to find $\gamma$ such that, $$\begin{aligned} \frac{1}{N} &=I\left(\gamma^2;\ \tfrac14,\tfrac14\right)\\[1.8mm] &= […]