Discrete dynamics of Eratosthenes sieve
Abstract
We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps π’(p#) of length Ο(p#) and span p#. There is a recursion π’(pk#)βΆ π’(pk+1#) that creates the next cycle from the current one. If we take initial conditions from the cycle π’(p0#), then for all constellations of span |s| < 2p1, including gaps g < 2p1, the driving terms of various lengths form Markov chains. These yield {\em exact} models for the populations ns(pk#) for all further stages of the sieve. If s is an admissible constellation of length J, then its population ns,J(p#) grows as Ξ( β (qβJβ1)). So we factor out the superexponential growth to obtain the exact model for the relative population ws(pk#) of the constellation s across all further stages of the sieve. ws,J(pk#) = ns,J(pk#) / βJ+1 < p β€ pk (pβJβ1) The asymptotic value of the relative population is a constant ws,J(β) β₯ 1 that depends only on the odd prime factors that divide a span in s. Assuming that the instances of a constellation s are approximately uniformly distributed in π’(pk#), we develop initial estimates of the number of instances s that would occur in the interval of survival ΞH(pk) = (pk2, pk+12). We define a statistic Ξ·s(pk), the quadratic density of the constellation s over the interval ΞH(pk). We show that the initial estimates Ξ·gΜ(p) for prime gaps agree with samples up to 5.677 E14.
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BibTeX
@misc{holt2026discrete,
title = {{Discrete dynamics of Eratosthenes sieve}},
author = {Fred B. Holt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.26384},
eprint = {2608.26384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26384},
}
RIS
TY - UNPB
TI - Discrete dynamics of Eratosthenes sieve
AU - Fred B. Holt
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2608.26384
UR - https://arxiv.org/abs/2608.26384
AB - We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps π’(p^#) of length Ο(p^#) and span p^#. There is a recursion π’(p_k^#)βΆ π’(p_k+1^#) that creates the next cycle from the current one. If we take initial conditions from the cycle π’(p_0^#), then for all constellations of span |s| < 2p_1, including gaps g < 2p_1, the driving terms of various lengths form Markov chains. These yield {\em exact} models for the populations n_s(p_k^#) for all further stages of the sieve. If s is an admissible constellation of length J, then its population n_s,J(p^#) grows as Ξ( β (qβJβ1)). So we factor out the superexponential growth to obtain the exact model for the relative population w_s(p_k^#) of the constellation s across all further stages of the sieve. w_s,J(p_k^#) = n_s,J(p_k^#) / β_J+1 < p β€ p_k (pβJβ1) The asymptotic value of the relative population is a constant w_s,J(β) β₯ 1 that depends only on the odd prime factors that divide a span in s. Assuming that the instances of a constellation s are approximately uniformly distributed in π’(p_k^#), we develop initial estimates of the number of instances s that would occur in the interval of survival ΞH(p_k) = (p_k^2, p_k+1^2). We define a statistic Ξ·_s(p_k), the quadratic density of the constellation s over the interval ΞH(p_k). We show that the initial estimates Ξ·_gΜ(p) for prime gaps agree with samples up to 5.677 E14.
ER -