[iclelland]

The overall density of primes is known, and decreases continuously as the numbers get larger (See http://en.wikipedia.org/wiki/Prime_number_theorem)

What this theorem says is that no matter how large the numbers get, no matter how sparsely the primes are spread out, on average, there will always be pairs of primes that are relatively close (within 600 of each other)

[ars]

"on average" is the wrong concept here. Just leave it out of your sentence and it's correct.

[thedufer]

That's what I thought, but if you apply "on average" to the clause preceding it rather than the one following it, it is correct.