Key of Primes Structure

Except  2 and 3

all  prime  numbers  have  values  equal

P = 6m ±1         ( m = 1, 2, 3…)

(P=6n+1 or 6n+5, “Primzahlen” von Dr. Ernst Trost, 1953)

2 and 3 are not like the rest of prime numbers;

they must be considered special, basic, primes.

For all primes except basic ones – 2 and 3

There are:

Classes of primes, gaps, and variations

For P

2 = {~ 0} = { +, - }
{
K~0 } = {+1, -1}

For V, G and g

3 = {k} = { 0, { ~ 0 }} = { 0, +, - }
{K k } = { 0, { K~0 }}= { 0,+1, -1}

Primes

 P+ = 6m + 1 P+mod 6 = 1 P- = 6m – 1 P-mod 6 = 5 Pk = 6m + K k

Gaps

Gmin = gmin = 2

 Pregaps Postgaps mod 6 For the kind  of pairs  of the consecutive primes G = 2m g = 2m {0,2,4} G0 = 6m g0 = 6m 0 sexy G+ = 2[ 3(m-1) + 1] g+ = 2 [ 3(m-1) + 1] 2 twin G- = 2( 3m - 1 ) g- = 2( 3m - 1 ) 4 cousin Gk = 2( 3m + K k) gk = 2( 3m + K k) 2K k

Variations

 V = 2a mod 6 ( a = 0, ± 1, ± 2, ± 3…) {0,2,4} V 0 = 6a 0 V + = 2 ( 3 a + 1 ) 2 V - = 2 = 2 ( 3 a -  1 ) 4 V k = 2 ( 3a + K k) 2K k

6666666666666666666666666666666666666666666666666666666666666666

There is another way to express primes greater than 3

on the base of 3       p = 3*(2m-1) ± 2

(1*6 = 1*3*2 )

See also the system of classes on the base of 3

Up ] Introduction of Variation ] Regular and Repeating Groups ] Groups in the Range until 100,000 ] Groups in the Range until 1,000,000 ] [ Key of Primes Structure ] Distribution of gaps and variations ] Boolean Algebra of Classes ] Consecutive primes ] Groups of primes in the infinite set ] Groups of primes in a limited range ] We have for each kind of regular and repeating groups ]

 Last updated 03/27/2009 Copyright © 2003 Michael Chassis. All rights reserved.