Is there any way to estimate the first number in the prime gap sequence which is greater than a given number $N$?
For example, for $N=3$, $g_4=11-7=4$ is the first one larger than $N$.
Thanks in advance.
2026-03-26 16:57:20.1774544240
The first number larger than $N$ in prime gap sequence
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You can stand on the shoulders of a giant, here.
There is a large prime gap between $n!+2$ and $n!+n$, this is completely elementary.
Erdos and Rankin used sieve methods to improve such bound, and the mentioned article by Maynard is a further improvement. In any case, this is a classical and pretty difficult problem, far from being completely settled.