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.

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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

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