# What theorem is this called? For any gap size n, no more than n gaps

of size n can consecutively occur in the sequence of primes.

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WWGD
Gold Member
2019 Award
Do you mean size exactly n? You can have gaps between primes that are as large as you want them to be.

Oops, I meant "occur in the sequence of prime gaps" not "occur in the sequence of primes", of course

e.g. for the gap size 12, no more than 12 gaps of size 12 can consecutively occur in the sequence of prime gaps 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, ....

Are you looking for something like this: http://arxiv.org/pdf/math/0508185v1.pdf
Upon reading over their intro, I would say it's similar, but not quite. I claim that, for any gap size n, at most n gaps of size n can consecutively occur in the sequence of prime gaps. The provided paper is an attempt at proving the Twin Primes Conjecture. I suppose my claim can be very easily proven and doesn't amount to much significance other than possibly getting school kids excited about learning remainders. :tongue2: For instance, it can be easily seen from remainders after dividing by 3 that the primes 3, 5, 7 produce the only instance of 2 gaps of size 2 appearing consecutively in the sequence of prime gaps.

Much thanks for the link. I recently came across the GPY result while reading about Zhang's work on the Twin Primes Conjecture.

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