Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png 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 […]

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: […]

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of “convergents constants” (defined at https://oeis.org/wiki/Convergents_constant) 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. […]

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 $\sqrt{e}$? 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 […]

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 […]

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 […]

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 […]

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.

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] &= […]

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