Is $\exists x,\ a_1, ... a_{n}$ such that $x^{n} = \sum_{i = 1}^{n} a_{i}^n$ true for any positive integer $n$?
Fermat's last theorem got me thinking if it's possible to split a hypercube of dimension $n$ with the length of each side being an integer into $n$ smaller hypercubes whose sides are also integer-valued, at least in principle. This is because of the way Fermat explained his theorem (that it's impossible to split a cube or any other higher power into two other cubes).
So, can you split a hypercube of dimension $n$ with integer sides into $n$ other smaller hypercubes also having integer sides? I'm not a mathematician, I'm just curious.