5 research outputs found
On a Dynamical Approach to Some Prime Number Sequences
In this paper we show how the cross-disciplinary transfer of techniques from
Dynamical Systems Theory to Number Theory can be a fruitful avenue for
research. We illustrate this idea by exploring from a nonlinear and symbolic
dynamics viewpoint certain patterns emerging in some residue sequences
generated from the prime number sequence. We show that the sequence formed by
the residues of the primes modulo are maximally chaotic and, while lacking
forbidden patterns, display a non-trivial spectrum of Renyi entropies which
suggest that every block of size , while admissible, occurs with different
probability. This non-uniform distribution of blocks for contrasts
Dirichlet's theorem that guarantees equiprobability for . We then explore
in a similar fashion the sequence of prime gap residues. This sequence is again
chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we
find forbidden patterns for every block of size . We relate the onset of
these forbidden patterns with the divisibility properties of integers, and
estimate the densities of gap block residues via Hardy-Littlewood -tuple
conjecture. We use this estimation to argue that the amount of admissible
blocks is non-uniformly distributed, what supports the fact that the spectrum
of Renyi entropies is again non-trivial in this case. We complete our analysis
by applying the Chaos Game to these symbolic sequences, and comparing the IFS
attractors found for the experimental sequences with appropriate null models.Comment: 18 pages, 20 figure