  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.
