User blog comment:Pellucidar12/Idea for a function (Hyperprime counting functions)/@comment-30754445-20170723221034/@comment-30754445-20170723224925

It's pretty trivial to see that there are infinitely many hyperprimes. Rigorously proving it may be difficult, but seeing it intuitively is easy. The primes are dense enough for there to be many many primes with a given sum-of-digits (unless the said sum of digits is divisible by 3, of-course).

Take 317 for example. There are may trillions of numbers whose sum of digits is 317 and about 1% (3/(2*317/4.5*ln(10))) of them would be prime.

Actually proving this could be a nighmare, though. Number theory is full with obvious statements which seem impossible to prove.