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Old 2018-05-13, 23:37   #2
ATH
Einyen
 
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Dec 2003
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You take the length of the gap call it: gn and divide it by ln(pn) where ln is the natural logarithm. The result is called the merit of the gap:
https://en.wikipedia.org/wiki/Prime_gap

merit = gn / ln(pn)

If you get merit > 30 you have a very large gap, if you get merit above 41.938784 you found a record gap.

It is proven that merit can be arbitrarily large as n gets large, so there should be much higher merits our there somewhere.


You can also divide the gap by ln(pn) twice:
gn / ln(pn)2

this is called the Cramér–Shanks–Granville ratio and the record is: 0.9206386 but anything above 0.5 is pretty good if your gap>1500.

The Cramér–Shanks–Granville ratio does not get arbitrarily large, but is conjected to have a maximum somewhere around 1.12

Last fiddled with by ATH on 2018-05-13 at 23:45
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