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); ...)


Schematic algebra