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

• I
• timmdeeg
In summary, the conversation discusses the discovery of a formula, (p-1)/(p-2), which predicts the frequency of prime number gaps when computing the first million prime numbers. The formula was found to be accurate within 1% and was explained by Professor Oskar Herrmann. The formula is not well-known among mathematicians and its relevance is questioned. Examples of how the formula yields the frequency of prime number gaps are also provided.

#### timmdeeg

Gold Member
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?

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

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?

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