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)
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
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); ...)