Recent content by andarrr

This follows easily from Minkowski's theorem on convex bodies: Given a lattice L in R^n with a fundamental parallelotope of volume V, then any convex, symmetric region in R^n with volume >4V contains a nonzero point of the lattice.

Catalan's theorem says that (1,1) and (2,3) are the only solutions to the first one, but I do not know if there is an elementary way of seeing this. As for the second problem, you can rewrite it as a^3=b^2+1, then factor over \mathbb{Z}[i] to get a^3=(b+i)(b-i). You can check that b+i and...

haha, I should have noticed that :redface:. Perhaps the original poster meant e^{2\pi i/p}, which would make the question slightly more interesting. edit: or even better, what Hurkyl said.

Maybe this is not in the ballpark of what you're looking for, but I believe you can approximate this using partial/Abel summation. We can approximate \pi(N)=\sum_{p<N} 1 and use this to approximate f(x): In particular, f(x) can be written as the Riemann-Stieltjes Integral f(x)=\int_{1}^{N}...