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K. Arthur
guest
Jul-31-05, 02:53 PM (EST)

"prime gaps"

 A005250 in NJAS' encyclopedia lists the increasing gaps betwen primes. There is another interesting sequence, A063095, which is defined asa(n) = max{ p(j+1) - p(j); j = 1,...,n } where p(j) is the jth prime,ie the largest difference between two consecutive primes in the sequence of the first n primes.Is there any known method to estimate the values of this function, without having to calculate it term by term? Even an approximation would be quite useful.

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Subject     Author     Message Date     ID
prime gaps K. Arthur Jul-31-05 TOP
RE: prime gaps mr_homm Jul-31-05 1
RE: prime gaps K.Arthur Aug-01-05 2
RE: prime gaps mr_homm Aug-01-05 3
RE: prime gaps K.Arthur Aug-02-05 4
RE: prime gaps alexb Aug-02-05 5
RE: prime gaps mr_homm Aug-02-05 6

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mr_homm
Member since May-22-05
Jul-31-05, 07:34 AM (EST)

1. "RE: prime gaps"
In response to message #0

 >Is there any known method to estimate the values of this >function, without having to calculate it term by term? Even >an approximation would be quite useful. Well, there is a theorem about the existence of gaps between primes. Simply, if x = 2*3*5*7*11*...*p, then x+1 and x-1 might be prime, but x+2, ..., x+p and x-p, ..., x-2 are not prime. So there is a gap of length at least p-1 near x. Therefore, a(n)>p-1, where n is the number of primes < x. This is a really weak result. Does this help at all?--Stuart Anderson

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K.Arthur
guest
Aug-01-05, 01:39 PM (EST)

2. "RE: prime gaps"
In response to message #1

 It'says something about the minimum of the largest gap within a range, but nothing about the maximum. Is it right to say that the Goldbach conjecture ( https://mathworld.wolfram.com/GoldbachConjecture.html ) cannot be proved if a stronger theorem cannot be found? What I mean is this:If u is a prime <= p, then p + u <= 2p. If v is a prime <= 2p + 1, then 2p + 1 + v >= 2p + 3.Therefore, if the gap following prime p is larger than p, then 2p+2 cannot be expressed as the sum of two primes.If that is right then, if the Goldbach conjecture is right, the following statement should also hold:The gap following prime p is not larger than p.Is this just another conjecture, or can it be (or has it been) proved?

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mr_homm
Member since May-22-05
Aug-01-05, 03:43 PM (EST)

3. "RE: prime gaps"
In response to message #2

 >The gap following prime p is not larger than p. >>Is this just another conjecture, or can it be (or has it >been) proved? Actually, this one is pretty obvious if you think about it from a very elementary point of view. 2p is obviously not prime, and lies a distance p above prime p. Hence the gap after p cannot exceed p.--Stuart Anderson

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K.Arthur
guest
Aug-02-05, 02:50 PM (EST)

4. "RE: prime gaps"
In response to message #3

 I do not understand why there has to be another prime between p and 2p.

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alexb
Charter Member
1610 posts
Aug-02-05, 02:52 PM (EST)

5. "RE: prime gaps"
In response to message #4

 Does not Bertrand's postulate answer this questionhttps://mathworld.wolfram.com/BertrandsPostulate.html?

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mr_homm
Member since May-22-05
Aug-02-05, 08:00 PM (EST)

6. "RE: prime gaps"
In response to message #4

 >I do not understand why there has to be another prime >between p and 2p. Sorry, I completely misread what you wanted. What alexb says in post #5 in this thread is what you want.--Stuart Anderson

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