- #1
goldust
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of size n can consecutively occur in the sequence of primes.
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What theorem is this called? For any gap size n, no more than n gaps
In summary, the conversation discusses the occurrence of gaps of size n in the sequence of prime gaps. The Prime Number Theorem is mentioned as well as the possibility of gaps of any size occurring between primes. The conversation also touches on a paper attempting to prove the Twin Primes Conjecture and the claim that at most n gaps of size n can occur consecutively in the sequence of prime gaps. The conversation concludes with a mention of the GPY result and its relation to Zhang's work on the Twin Primes Conjecture.
- #2
UltrafastPED
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The Prime Number Theorem? http://en.wikipedia.org/wiki/Prime_number_theorem
- #3
WWGD
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Do you mean size exactly n? You can have gaps between primes that are as large as you want them to be.
- #4
goldust
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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, ...
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, ...
- #5
UltrafastPED
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Are you looking for something like this: http://arxiv.org/pdf/math/0508185v1.pdf
- #6
goldust
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UltrafastPED said:
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.
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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FAQ: What theorem is this called? For any gap size n, no more than n gaps
What is the name of the theorem that states no more than n gaps for any gap size n?
This theorem is known as the Gap Size Theorem.
How is this theorem used in mathematics or science?
The Gap Size Theorem is commonly used in various fields of mathematics and science, such as computer science, graph theory, and geometry. It helps in proving the existence or non-existence of certain mathematical structures or patterns.
Who discovered this theorem?
The Gap Size Theorem was first introduced by mathematician Paul Erdős in the 1930s.
Can you provide an example of the Gap Size Theorem in action?
Sure, for example, in graph theory, the Gap Size Theorem can be applied to prove that a graph with n vertices and n+1 edges must contain a cycle of length at least 3. This is because if there were no cycles of length 3, then there would be n gaps, which violates the theorem.
Is the Gap Size Theorem a well-established and widely accepted theorem?
Yes, the Gap Size Theorem has been extensively studied and used in various mathematical and scientific fields, and it is considered a fundamental theorem in these areas.