﻿ Distribution of primes in G(n,k)

## Distribution of primes in G(n,k)

#### E-classes

Classes, whose g-number is not divisible with r = n^k-1, are so called E-classes. We expect, there would be a higher density of prime numbers in E-classes than in other classes.
Number of E-classes in G(n,k) is:

`    φ(n^k-1)/k`

The less E-classes a G-system has, the higher concentration of primes we expect in these E-classes.

Let P(r) be the number of primes p
Therefore

`  P(r) <= φ(r)+ F(r)`

#### Primes in G(2,k)

Example of prime numbers in the class g=n^(k-1) of systems G(n,k):
(prime numbers are marked by asterisk)

` k |    r  |  1    2     4     8     16    32    64    128`
`---+-------+-------------------------------------------------`
` 3 |    7  |  6   *5    *3`
` 4 |   15  | 14  *13   *11    *7`
` 5 |   31  | 30  *29    27   *23     15`
` 6 |   63  | 62  *61   *59    55    *47   *31`
` 7 |  127  |126  125   123   119   *101    95    63`
` 8 |  255  |254  253  *251   247   *239  *223  *191   *127`

#### Primes in G(n,2)

In the following tables some ratios are tested in systems G(n,2):

 G( n,2) r φ(r) φ(r)/k F(r) P(r) E(r) G( 2,2) 3 2 1 0 2 2 G( 3,2) 8 4 2 1 5 4 G( 4,2) 15 8 4 2 7 5 G( 5,2) 24 8 4 2 10 8 G( 6,2) 35 24 12 2 12 11 G( 7,2) 48 16 8 2 16 14 G( 8,2) 63 36 18 2 19 17

 G( n,2) E(r)/P(r) E(r)/φ(r) [E(r)+F(r)]/[φ(r)+F(r)] P(r)/[φ(r)+F(r)] G( 2,2) 1.00 1.00 1.00 1.00 G( 3,2) 0.80 1.00 1.00 1.00 G( 4,2) 0.71 0.63 0.70 0.70 G( 5,2) 0.80 1.00 1.00 1.00 G( 6,2) 0.92 0.46 0.50 0.46 G( 7,2) 0.88 0.88 0.89 0.89 G( 8,2) 0.89 0.47 0.50 0.50

Prime pairs

The following example brings classes containing prime pairs [u(0),u(1)].
Classes are sorted by number s, s = (∑u(j))*(n-1)/(n^k-1).

`  G( 2,2) 1: [ 2, 1]`
`  G( 3,2) 1: [ 3, 1]     3: [ 7, 5]`
`  G( 4,2) 4: [13, 7]`
`  G( 5,2) 1: [ 5, 1]     3: [11, 7]`
`          5: [17,13]     7: [23,19]`
`  G( 6,2) 6: [31,11]`
`  G( 7,2) 1: [ 7, 1]`
`          5: [23,17], [29,11]`
`          7: [37,19], [43,13]`
`         11: [47,41]`
`  G( 8,2) 6: [41,13]     8: [43,29]`
`         10: [59,31]    12: [61,47]`
`  G( 9,2) 3: [19,11]     5: [37,13]    7: [47,23]`
`          9: [73,17]    11: [67,43]   15: [79,71]`
`  G(10,2) 4: [31,13]     8: [71,17]`
`         10: [73,37]    16: [97,79]`
`  G(11,2) 1: [11, 1]`
`          3: [23,13]`
`          7: [ 47,37], [ 67,17]`
`          9: [ 79,29], [ 89,19]`
`         11: [101,31], [ 71,61]`
`         13: [ 83,73], [103,53], [113,43]`
`         17: [107,97]`
`  G(12,2) 6: [61,17]    12: [89,67]`

Prime pairs [u(0),u(1)] in E-classes has distance d(i). Each distance is divisible by (n-1).
In the following table values d(i)/(n-1) for particular prime pairs are listed.

`               d(i)/(n-1)`
` -----------------------------------------`
`  n = 2:       1`
`  n = 3:       1,1`
`  n = 4:       2`
`  n = 5:       1,1,1,1`
`  n = 6:       4`
`  n = 7:       1,1,1,3,3,5`
`  n = 8:       2,2,4,4`
`  n = 9:       1,1,3,3,3,7`
`  n =10:       2,2,4,6`
`  n =11:       1,1,1,1,1,1,5,5,5,7,7,7`
`  n =12:       2,4`

Conjecture

For each positive integer n>2 there exist primes p, q satisfying:
p+q = n(n+1)
Example:

`  n = 3:     5+  7         =  3*4`
`  n = 4:     7+ 13         =  4*5`
`  n = 5:    13+ 17         =  5*6`
`  n = 6:    11+ 31         =  6*7`
`  n = 7:    13+ 43 = 19+37 =  7*8`
`  n = 8:    29+ 43         =  8*9`
`  n = 9:    17+ 73         =  9*10`
`  n =10:    37+ 73         = 10*11`
`  n =11:    31+101 = 61+71 = 11*12`
`  n =12:    89+67          = 12*13`

Dual primes

Primes are dual according to module m if m\(p*q-1), i.e. p*q = 1 (mod m).
Example

`  5: (2,3)`
`  7: (3,5); (2,11); (5,17); (29); (43); (57);`
` 11: (2,17);`
` 13: (2,7); (5,8);`
` 17: (5,7);`
` 19: (3,13);`
` 23: (3,31);`

(Various special cases might be defined, e.g.: p+q=m: 5: (2,3) 13: (5,8); ...)