Statistical Bias in the Distribution of Prime Pairs and Isolated Primes

TL;DR

This paper uncovers a previously unknown bias in prime distribution, showing that twin primes and isolated primes are more likely to be centered on nonsquarefree multiples of 6 than expected, with implications for understanding prime patterns.

Contribution

It reveals a novel statistical bias in prime distribution related to nonsquarefree multiples of 6, supported by extensive computational data up to 10^10 primes.

Findings

Twin primes are 6.0% more often centered on nonsquarefree multiples of 6 than expected.

Isolated primes show a 1.9% deviation from expected distribution.

Bias towards nonsquarefree numbers increases with larger prime counts.

Abstract

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.