The dynamic of prime numbers
0x6251
May 17th, 2022

A prime numbers sieve that unravels as periodic sequence of co-primes of an ever growing set of prime generator numbers that depicts the periodic distribution of prime numbers.

Fourier analysis and number theory article below suggest ”the existence of some kind of mysterious dynamical system underlying (or "lurking behind" as N. Snaith put it in her Ph.D. thesis) the distribution of prime numbers.”.

It’s my hope that this sieve can prove this dynamic from the inside out as it generates the set of primes by means of composite periodic sequences of probable-primes. As the set of generator prime numbers tends to infinity so does the sequence period.

Proposition

All prime numbers (ℙ) are contained in the sequence:

Sieve definition

Initial iterations

Implications

Direct observations of the behavior of the sequence Pn

Table 1: n=4; Pn Gn=[2,3,5,7]; Highlighted the symmetric gaps after removing multiples of 7 importantly {7, 203}
Table 1: n=4; Pn Gn=[2,3,5,7]; Highlighted the symmetric gaps after removing multiples of 7 importantly {7, 203}
Table 2: Tn separating last & first elements in each contiguous repetition of the sequence, generating twin primes.
Table 2: Tn separating last & first elements in each contiguous repetition of the sequence, generating twin primes.
Table 3: Progression of |P| & Tn in relation to π(x)
Table 3: Progression of |P| & Tn in relation to π(x)

Skipping |G| between 26 and 94:

Table 4: Progression of |P| & Tn in relation to π(x) |G| > 94
Table 4: Progression of |P| & Tn in relation to π(x) |G| > 94
Image 1: Black vertical Line Tn=6; Red Tn=30; Blue Tn=210
Image 1: Black vertical Line Tn=6; Red Tn=30; Blue Tn=210

Algorithmic approaches

See example below. This enables us to expand Pn to the limits of available RAM (and data structures), while continue to increase accuracy of the sequence’s primes locations prediction for very big numbers.

Optimized for performance

TODO.

Optimized for space

As |P| grows exponentially, in order to make the most use of RAM a BitMap representation is used to address as many bits as possible with the least overhead. The programing language of choice is Java and the data structure that most align with this is BitSet, java.util.BitSet although in it’s API it only allows for int values used as indexes, which limits the addresses to INTEGER_MAX_VALUE while the (implementation dependent) underlying data-store is long[]. In order to store the most number of bits, java.util.BitSet was modified to allow for long typed indexes, code for LongBitSeg is stored under my account on github.

A basic implementation of the sieve is published on github Here. Executing main() method in current revision produces, for |G| = 6:

 * Computing Co-Primes Finished in 6.762224 ms. 
   With Value: 
    => |P| = 5760 ; T(P) = 30030 ; [G] = [2, 3, 5, 7, 11, 13]

 * Eliminating non-primes from [P] (except 1) Finished in 2.878282 ms. 
   With Value: 
    => Incomplete Harmonic has 
  T(iP) = 166589903787325219380851695350896256250980509594874862046961683989710 = 1.665E+69
  Allows to evaluate primality of integers in intervals [ n × T(iP) ± 30030 ]
  With Generator primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173]

 * sieveOfEratosthenes Finished in 4.354895 ms.
Primes in sieveOfEratosthenes:3248
Primes in Harmonics Sieve:    3248

A version of sieve of Eratosthenes that also uses LongBitSet is used as precision benchmarking.

For |G| = 10 (current memory limit), output trimmed for space:

 * Computing Co-Primes Finished in 3282.685708 ms. 
   With Value: 
    => |P| = 1021870080 ; T(P) = 6469693230 ; [G] = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

 * Eliminating non-primes from [P] (except 1) Finished in 21541.873628 ms. 
   With Value: 
    => Incomplete Harmonic has 
  T(iP) = 28165847668170109...490685420551205570 = 2.816E+34777
  Allows to evaluate primality of integers in intervals [ n × T(iP) ± 6469693230 ]
  With Generator primes: [2, 3, 5, 7, 11, ..., 80407, 80429]

 * sieveOfEratosthenes Finished in 107445.191176 ms.
Primes in sieveOfEratosthenes:300369796
Primes in Harmonics Sieve:    300369796

This implies that with 10 generator primes [2, …, 29], we can generate an incomplete Pi sequence with period Ti = 2.816E+34,777 with much more accurate prediction of probable primes than Pn in the vicinity of:

For an accessible (though inefficient) Javascript implementation see here

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