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Old 2018-05-14, 14:35   #5
rudy235's Avatar
Jun 2015
Vallejo, CA/.

13·79 Posts

Originally Posted by Bobby Jacobs View Post
I would really like a measure of record prime gaps where a big value for a record gap between small numbers is a big value for a record gap between large numbers. The size of the gap will not work. For example, the prime gap between 1327 and 1361 has size 34, which is very big for numbers that size. However, a gap of 34 between larger numbers is not very great. The merit of a gap has the same problem. The gap from 1327 to 1361 has merit 4.7, but a merit of 4.7 is not impressive for larger numbers. Even the CSG ratio is not perfect. The CSG ratio of the gap from 1327 to 1361 is 0.65. However, the CSG ratios of the bigger record prime gaps are all at least 0.8. Do you know a prime gap measure where a good value of a record prime gap between small numbers is a good value of a record prime gap between large numbers?
OK once again.
The average gap between 2 primes is approximately ln(p1) -where p1 is the lowest end of the gap-.

This comes from the Prime Number Theorem (PNT) proven by Hadamard and de la Vallée Poussin in the late XIX Century

A gap can in theory be as small as 2 and as big as around ln2p1

Having said that the "merit" which is the ratio of gn/ln p1 gives an excelent idea of how big is the gap in terms of probability. Merits of 1 are the most common. Merits of 0.5 or of 2 are less frequent. Merits of 0,1 or 10 are even less common and the largest (or smallest) a merit is -relative to 1- the less common it is. It follows more or less a Gaussian distribution with the peak at ln(p).

However (as it has been said before) merits can also be larger than any preset number, if we accept the Cramér's conjecture For instance in the veciniry of a Googolplex 10^(10^100) you should be able to find merits as large as 530
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