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.
All prime numbers (ℙ) are contained in the sequence:
|G| between 26 and 94:
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.
|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.
* 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.
|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: