I Frequency of prime number gaps according to (p-1)/(p-2)

[timmdeeg]

TL;DR Summary

I wonder why this formula seems to be widely unknown. I can't find it in the web. "The New Book of Prime Number Records" says something about prime gaps without mentioning this formula though.

Caution I'm not a mathematician. In short, long time ago I calculated prime number gaps just for fun expecting an almost uniform distribution of the frequency of the gaps 2, 4, 6, ... . Instead the frequency showed a series of maxima and minima and I was confused. Later Professor emeritus Oskar Herrmann University Heidelberg cleared my question up and explained (p-1)/(p-2) which has been proved heuristically by Polya and Lehmer the first half of the 20th century. I have that in German. The prediction of this formula confirmed my results within about 1%.

Perhaps this is too trivial to be of interest for mathematicians. What is your opinion?

 

[Office_Shredder]

Sorry, can you be a bit more specific what this formula is supposed to say about prime gaps?

 

[timmdeeg]

Hm perhaps "frequency" and "gap" aren't the correct expressions.

My program has computed the first million prime numbers. From that I obtained how often the difference between any two prime numbers is 2, how often 4, 6 and so on. I call this number N(i) whereby i = 2, 4, 6, ...
From this (p-1)/(p-2) yields a probability of N(i) relativ to N(2):

Example

Difference .......... (p-1)/(p-2)

2 ............. 1.00
4 ............. 1.00
6 ............. 2.00
8 ............. 1.00
10 .......... 1.33
12 .......... 2.00
14 .......... 1.20
16 .......... 1.00
18 .......... 2.00
.
30 .......... 2.66
.
42 .......... 2.40
.
210 ....... 3.20

Hope that is more clear now, I can also show examples how to get those figures from (p-1)/(p-2).

Differenz 6 : 2*3 : (3-1)/(3-2) = 2
Differenz 10: 2*5 : (5-1)/(5-2) = 1.33
Differenz 30: 2*3*5 : [(3-1)/(3-2)]*[(5-1)/(5-2)] = 2.66

Is (p-1)/(p-2) widely unknown or just not of any relevance?