Number Theory

   

Statistical Bias in the Distribution of Prime Pairs and Isolated Primes

Authors: Waldemar Puszkarz

Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $\pi^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towards nonsquarefree numbers, a phenomenon most likely previously unknown. For twins, this leads to nonsquarefree numbers gaining an excess of $1.2\%$ of the total number of twins. In the case of isolated primes, this excess for nonsquarefree numbers amounts to $0.4\%$ of the total number of such primes. The above numbers are for the first $10^{10}$ primes, with the bias showing a tendency to grow, at least for isolated primes.

Comments: 7 Pages. Last version submitted to viXra. First version submitted to arXiv.

Download: PDF

Submission history

[v1] 2018-04-27 20:31:50
[v2] 2018-05-01 15:11:21
[v3] 2018-05-03 22:03:38
[v4] 2018-05-05 18:14:28
[v5] 2018-05-15 17:05:10
[v6] 2018-05-26 16:55:29
[v7] 2018-06-27 19:48:55
[v8] 2018-07-02 20:34:06

Unique-IP document downloads: 37 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus