Patterns among the Primes
A study of Eratosthenes Sieve
Welcome! This is a study in discrete dynamic systems, in which the object of study is Eratosthenes sieve.
Specifically the cycles of gaps G(p#) at each stage of the sieve.
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At each stage of Eratosthenes sieve there is a corresponding cycle of gaps G(p#) between the remaining candidate primes.
There is a recursive construction that produces the next cycle from the previous one.
This recursion across the cycles of gaps G(p#) is a discrete dynamic system.
Within this discrete dynamic system we are able to identify exact population models for gaps and for constellations of gaps.
p# is the primorial of p,
the product of all the primes
up to and including p.
Just as the prime numbers are candidate primes that survive the sieve, the gaps between primes are the gaps in G(p#) that survive. Every gap between primes is a gap that arose in the cycles G(p#) and survived.​
We are able to create exact population models for small gaps,
across the stages of Eratosthenes sieve.
These models fairly predict the distributions of the gaps that survive the sieve to become gaps among primes.
The cycles of gaps G(p#)
When a candidate prime is removed by the sieve, the gaps on either side are fused, added together.
Interestingly, Alphonse de Polignac identified these cycles of gaps in his 1849 paper in which he stated the conjecture that bears his name.
Polignac's Conjecture. Every number g=2k occurs infinitely often as a gap between consecutive prime numbers.
There is a recursive construction that produces the next cycle of gaps from the current one. To create G(7#) from G(5#), we concatenate 7 copies of G(5#) and then we add together adjacent gaps - fusing the gaps - as indicated by the element-wise product 7*G(5#).
G(7#) consists of 48 gaps and has period 210=7#.
Many patterns are preserved from one cycle of gaps G(p#) to the next. The dynamic system tracks these patterns.