It seems from a quick check that for all $m$, there is at least one $n$ where $m>n>0$ and $m^2+n^2\in\mathbb P$.
Is this true, and is there a proof?
Empirically, it looks likely to be true, and the number of primes per given $m$ climbs fast. Furthermore, if it's provable that there are at least two solutions for any $m>12$ (which seems very likely heuristically), an alternative proof of Bertrand's postulate would immediately follow:
If we're guaranteed two prime pairs per $m$, the largest possible difference between two consecutive $m$ pairs changes from
Then we have
and that should do it.