|
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
|
|
P k
= 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 ]
|
|