As a fun little exercise I wrote an algorithm that will give a
minimum bound for pi(p(i)^2).
I wish I could turn at least part of it into some sort of proof.
(For example, for all i>12, can it be proved that there are at least
X many primes between p(i)^2 and p(i)?) But I'm not skilled at
proofs, so the paper only highlights an algorithm.
(Is there such a proof out there with that form, by the way? I tried
to ask about that earlier, but I think my email got drowned in a
flurry of other posts. Or maybe I asked the question poorly.)
It's probably nothing significant... just a little trinket. But I
wrote it up as I explored it if anyone wants some light reading:
Hope everyone who's on holiday is having a pleasant and safe time.