I found two consecutive prime numbers $401!-3463$ and $401!+4021$. These have a difference of $7664$. Is there some kind of technique that is known in order to find consecutive prime numbers with sufficiently large gaps?
I just used the fact that for all positive integers $n$, it follows that all the integers on the interval $[n!+2, n!+n]$ are composite, which can make great prime gaps. Are there other more efficient techniques?
Apologies if this question is too broad. I was just curious :)