# Infinitely many maximal and nonmaximal prime gaps?

This is a simple question about first occurrence prime gaps and maximal and nonmaximal prime gaps.

A gap between prime numbers is maximal if it is larger than all gaps between smaller primes.

My questions are: is there is a proof that there are infinitely many maximal prime gaps? Is there is a proof that there are infinitely many nonmaximal prime gaps?

• Since there are arbitrarily large gaps - $N! + 2,\, \dotsc,\, N! + N$ - it follows that there are infinitely many maximal gaps. – Daniel Fischer Aug 10 '13 at 22:40
There are arbitrarily large prime gaps (all the numbers $N! + k$ for $2 \leqslant k \leqslant N$ are composite), hence there are infinitely many maximal prime gaps.
$$\pi(x) \sim \frac{x}{\log x}.$$