I've reported on this in the distant past, but since Mike Lawler recently asked bloggers to post some entries that might be of interest to both mathematicians and students, I’ll re-run this simple, old demonstration that you can have ANY size gap between two prime numbers that you want. I’ve always liked it, for its simplicity, since first seeing it in a popular 1984 volume from Laurie Buxton called “Mathematics For Everyone.” It runs like this (using Buxton’s example):
Hopefully you know what 600! means, i.e. the product of 600 x 599 x 598 x ….. x 2 x 1.
A pretty large number, but we need not actually multiply it out. Now consider the following string of consecutive numbers:
600! + 2
600! + 3
600! + 4
600! + 5
600! + 600