Sum of two $n$-th powers, of two $m$-th powers, but not of two $mn$-th powers Let $m$ and $n$ be two coprime numbers. Let $S_n$ denote the set of integers that are a sum of two $n$-th powers of integers, for example $7\in S_3$ given $7=2^3+(-1)^3$. Analogously define $S_m$ and $S_{mn}$.
My question is if this set is infinite: $(S_n\cap S_m) \smallsetminus S_{mn}$. 
For $n=2$ and $m=3$ I could manage to prove it is infinite and the solution is elementary (constructive). However, I don't see any clue of how to prove that this is true for arbitrary $n$ and $m$ (obviously, here we are allowed to use as much machinery as needed).
 A: Likely it is not known how to answer your question in general, but here are some comments that perhaps are useful. 
Note that $S_2$ is massive. If we put $S_2(X) = \{x \in S_2: x \leq X\}$, then $S_2(X) \sim c_0 X (\log X)^{-1/2}$, where $c_0$ is Landau's constant. Thus it is very easy for $S_2(X)$ to ``accidentally" contain elements of $S_3(X)$, which has density $\sim c_1 X^{2/3}$ for some explicit constant $c_1$ (this was first computed in a paper by T. Wooley). Note that $S_6(X)$ has density $c_2 X^{1/3}$ (see https://link.springer.com/article/10.1007%2Fs00208-019-01855-y) for some $c_2 > 0$, so heuristically most of these occurrences will not be accounted for by the second phenomenon. 
In general, for each $m$ there exists a constant $c_m$ such that $S_m(X) \sim c_m X^{2/m}$. When $m,n$ are both large, i.e. when $\frac{2}{m} + \frac{2}{n} < 1$, then it is unlikely that $S_m(X), S_n(X)$ intersect at all, unless there is a reason for them to. The reason of course is that $S_{mn}(X)$ is contained in both. 
Another comment is that when $\min\{m,n\} \geq 5$ the surface $x^m + y^m = u^n + v^n$ is of general type, so conjecturally the set of rational points is not Zariski dense. Since there is an obvious subvariety containing lots of points (corresponding to $S_{mn} \subset S_m \cap S_n$), the naive conjecture is that this subvariety contains all of the points. 
