From: scerir (scerir@libero.it)
Date: Thu Mar 27 2003 - 15:04:00 MST
Information Entropy and Correlations in Prime Numbers
http://xxx.lanl.gov/abs/cond-mat/0303110
Pradeep Kumar, Plamen Ch.Ivanov, H. Eugene Stanley
14 pages, 6 figures
The difference between two consecutive prime numbers is called the
distance between primes. We study the statistical properties of the
distances and their increments (the difference between two consecutive
distances) for a sequence comprising the first $5\times 10^7$ prime numbers.
We observe that the sequence of distances and the sequence of increment
magnitudes both exhibit remarkable logarithmic trends. We obtain an
empirical form for the information entropy for the set of distances as a
function of sequence length $N_p$, and a very similar form for the
information entropy for the set of corresponding increments. We find that
for a given $N_p$, the entropy of the set of increments is always greater
than the entropy of the set of distances, suggesting a greater variability
in the values of the increments. We also find that the histograms of the
distances and the increments follow exponential distributions with
superposed periodic behavior with different periods: period-three
oscillations for the magnitude of increments similar to previously reported
period-six oscillations for the distances. We further investigate the
correlations between the distances as well as their increments and find that
at small and intermediate scales the distances exhibit a weak power-law
anticorrelation followed by a crossover to strongly correlated behavior at
large scales. This crossover is due to the logarithmic trend in the sequence
of distances. For the increments, we find that (i) they are strongly
anticorrelated at all scales, (ii) their magnitudes exhibit a crossover from
weakly anticorrelated behavior at small scales to strongly correlated
behavior at large scales, and (iii) their signs are anticorrelated at small
scales and uncorrelated at large scales.
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