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K. Arthur
guest

Jul3105, 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 as a(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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mr_homm
Member since May2205

Jul3105, 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 x1 might be prime, but x+2, ..., x+p and xp, ..., x2 are not prime. So there is a gap of length at least p1 near x. Therefore, a(n)>p1, 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

Aug0105, 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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