Recently came across a blurb regarding factorials in the Appendix of Laurie Buxton's old book, "Mathematics For Everyone" which I thought was quite lovely and I didn't recall reading before. I'll just quote verbatim (I've added bold):
"Now consider the string of consecutive numbers:To me, another simple yet beautiful mathematical thought process here... in fact, makes me wonder how much factorials have been used in approaching the Riemann Hypothesis, or more generally, the distribution of primes?
600! + 2
600! + 3
600! + 4
600! + 600
There are 599 consecutive numbers here, and none of them can be prime, for every one divides by at least the number on the right. For instance, 600! + 59 must divide by 59, since every number up to 600 divides into 600! This gives us a gap of 599 with no primes. We chose 600 at random to start. It is clear that we could start with a number as large as we like, and hence make the gap as large as we like."