A study of divergence from randomness in the distribution of prime numbers within the arithmetic progressions 1+6n and 5+6n

Abstract

In this article I present results from a statistical study of prime numbers that shows a behaviour that is not compatible with the thesis that they are distributed randomly. The analysis is based on studying two arithmetical progressions defined by the following polynomials: ($1+6n$, $5+6n$, $n\in{N}$) whose respective numerical sequences have the characteristic of containing all the prime numbers except $3$ and $2$. If prime numbers were distributed randomly, we would expect the two polynomials to generate the same number of primes. Instead, as the reported findings show, we note that the polynomial $5+6n$ tends to generate many more primes, and that this divergence grows progressively as more prime numbers are considered. A possible explanation for this phenomenon can be found by calculating the number of products that generate composite numbers which are expressible by the two polynomials. This analysis reveals that the number of products that generate composite numbers expressible by the polynomial $1+6n$ is $(n+1)^{ 2}$, while the number of products that generate composites expressible by the polynomial $5+6n$ is $(n+1)n$, con $n\in{N}$. As a composite number is a non-prime number, this difference incited me to analyse the distribution of prime numbers generated by the two polynomials. The results, based on studying the first (approx.) 500 million prime numbers, confirm the fact that the number of primes that can be written using the polynomial $1+6n$ is lower than the number of primes that can written using the polynomial $5+6n$, and that this divergence grows progressively with the number of primes considered.