The results
reported in this note refer to the distribution of z_{n}
= p_{n} - p_{n-1} for the first three million prime numbers (p). The analyses
of the note are almost purely statistical. The difference between consecutive
prime numbers is treated as a random variable, and empirical frequency
distributions are examined for sets of 5000 consecutive primes through
the first three million. The results reported are based upon frequency
distributions (59 in total) that are calculated at intervals of 50,000
primes for π(n) between 95,000 and 3,000,000. The quantities that
are investigated include the means and standard deviations of the 59
distributions, together with coefficients that are obtained from exponential
functions fitted by least-squares to the “poles” of the
underlying density functions. The resulting vector of 59 estimated coefficients
is then in turn related (via a least-squares regression equation) to
the logarithm of p_{n}.

Key results
of the analyses are as follows:

(1) That
the mean of z_{n} increases with the logarithm of p_{n} is clearly

confirmed.

(2) The
support for z_{n} increases very slowly through the first three

million primes, as the maximum z_{n} in the “samples” of 5000

consecutive primes that have been analyzed is never found to be

larger than 178.

(3) “Poles”
in the distribution of z_{n} are present at values of z_{n} divisible by

six. These “poles” have an analytical basis, and appear
to decline

exponentially.