I have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would like to study the arithmetical part related to imaginary quadratic fields and their relation to elliptic functions. Having very limited knowledge of […]

EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 – z^3 \neq \pm 2 m^3, \; \pm 32 m^3$ when $m$ has certain prime factorizations. The answers appeared in the […]

I’m curious, how one can prove the following integral $$ \int_0^{\pi/2}{\frac{1+2\cos 2x\cdot\ln\tan x}{1+\tan^{2\sqrt{2}} x}}\tan^{1/\sqrt{2}} x~dx=0 $$ Here is the Wolfram Alpha computation which shows that it is correct to at least 45 digits. My attempt: I knew the integral $$ \int_0^{\pi/2}\frac{1}{1+\tan^\alpha\phi}d\phi=\int_0^{\pi/4}\frac{1}{1+\tan^\alpha\phi}d\phi+\int_0^{\pi/4}\frac{\tan^\alpha\phi}{1+\tan^\alpha\phi}d\phi=\frac{\pi}{4} $$ which can be calculated for all values of $\alpha$. I tried to find […]

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