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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?

Thanks in advance!

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  • $\begingroup$ Since there are arbitrarily large gaps - $N! + 2,\, \dotsc,\, N! + N$ - it follows that there are infinitely many maximal gaps. $\endgroup$ – Daniel Fischer Aug 10 '13 at 22:40
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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.

If there were only finitely many non-maximal prime gaps, the sequence of prime gaps would eventually become strictly increasing, and that would mean that the sequence of primes grows at least quadratically in the long run, which contradicts the Prime Number Theorem

$$\pi(x) \sim \frac{x}{\log x}.$$

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