|
|
|
|
|Store|
|
|
|
|
CTK Exchange
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 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. |
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
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 |
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
 |
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?
|
|
|
Alert | IP |
Printer-friendly page |
Reply |
Reply With Quote | Top |
|
|
|
You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.
|Front page|
|Contents|
Copyright © 1996-2018 Alexander Bogomolny
|
|
Printer-friendly copy
Email this topic to a friend
