The maximal sizes of regular and irregular repeating groups
depend on special dynamic characteristics of each type of them.
In the range until 1,000,000,000, they may be calculated with the formula
Maximal Regular and Repeating Groups
The maximal sizes of regular and irregular repeating groups
depend on special dynamic characteristics of each type of them.
In the range until
1,000,000,000, they may be calculated with the formula
where
| Dynamic coefficient d | = 0 | if the variations in a group do not change their values |
| = 1 |
if they change their values |
|
| = -1 | if the signs of the consecutive variations alternate only | |
| Symmetry coefficient s | = 0 | if group is irregular |
|
= 1 |
if the variations symmetrical of the center of a regular group have the same sign (a group has polar symmetry) |
|
| = -1 |
if they have the opposite sign (a group has axial symmetry) |
|
|
For special types of regular groups, it must be taken |s| =
1 |
||
| Size coefficient s | = 1 | if the group is odd |
| = 2 | if it is even | |
|
If the type of the groups can have s = 1 or 2
without changing their main properties |
||
| Gaps coefficient g | = 1 | if the pregap of a prime is equal to its postgap (one zero variation creates a group of 2 equal consecutive gaps) |
| = 0 | if it is not |
|
Type |
d |
s |
g |
s |
Sizemax |
||
|
Odd |
Even |
Odd |
Even |
||||
| Symmetriads | 1 | -1 | 0 |
1 |
2 |
7 |
12 |
| Curls | 1 | 1 | 0 |
1 |
2 |
13 |
10 |
| Flats | 0 | 1 | 1 | 1 | 5 (number of gaps) |
| Zigzags | -1 | 1 | 0 | 1 | 6 |
| Stairs | 0 | 1 | 0 | 1 | 9 |
| Irregular repeating groups | 1 | 0 | 0 | 2 | 12 (in the range until 3,000,000) |
The fact that the maximal size of irregular repeating groups found in the range until 36,600
has not increased in the range until 3,000,000
gives possibility to suppose it will not increase in the range of primes until 1,000,000,000, too.
The maximal sizes of regular groups do not change their values from 243,333,233 until 1,000,000,000.
