Specific Question Let $R$ be a whole number. Is it possible for any whole number $c>0$ that there is a nicer form for $\sum_{r=1}^{R^c}\sum_{d|r}(-1)^{r+d} d^3$? I can hope that we can just evaluate this sum but in lieu of that maybe something cheaper computationally. Taking $c=1$ we can exami...

Consider a differential equation with a term containing $y(x_0)$, for example $$y'' - 2y' + y = y(x_0)$$ $x_0 \in \mathbb{R}$ is a constant. My question is, does such equations fall under the category of differential equations? I have never studied any equation with such a term. If its a differe...

Looking for a complete regular Riemannian metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic wrt. the metric, $x,y\in(0,1), s\in \Bbb R(1, \infty). $

A sufficiently large prime number $n$ can only be in certain residue classes modulo another integer $m$. For instance modulo $6$, these classes are $1, 5$. The number of classes precisely is $\varphi(n)$ or the value of Euler's totient function. Can we use this information in prime number prob...