No Largest Prime Gap

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

The above produces a list of 599 consecutive integers, NONE of which can be prime. Every number here will be divisible by at least the number on the right (because 600! is divisible, without remainder, by every number UP TO 600, and adding anything between 1 and 600 simply includes one of those divisors). Thus, in this example we have a gap of at least 599 integers without a prime appearing. BUT clearly one need not start with 600. One can start with a number as large as one likes in order to generate a prime gap as large as one wants. There will never be a largest gap. Simple and convincing!