Consider the following lacunary sum with parameter $x$: $$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$ As we can see for $x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\frac{π}{2p}\right)$$ where $p$ runs through all primes. What are some non trivial properties of $S...

Consider the following lacunary sum with parameter $x$: $$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$ As we can see for $x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\frac{π}{2p}\right)$$ where $p$ runs through all primes. What are some non trivial properties of $S...

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula $v-e+f=2$ to show that there are exactly five possible triples $(v,e,f)$. But of course this is not a ...

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula $v-e+f=2$ to show that there are exactly five possible triples $(v,e,f)$. But of course this is not a ...