The proof shows all numbers cycle through the six different cell types (n, n+1, etc.) as the numbers spiral about within the six hexagons. I’ve reclassified the cells per modulo 6, revealing a simplified version in figure 5. I’ve added colors to the classification of the cells for simpler identification.
The hexagon, modulo 6.
Some Characteristics of cells
2- cell: consists of only even numbers; but not even numbers divisible by 3. Odd powers of 2 reside in this cell.
3- cell: contains ALL numbers divisible by three (‘threeven’), unless they are even. Thus, any odd number (primes), multiplied by three, lands here, as do all powers of 3.
These two families border the hexagon’s outer edge to create the boundary conditions to confine the number line as the always contain composite numbers.
0- cell type: contains all numbers divisible by both 2 and 3, and, obviously all numbers evenly divisible by 6.
4- cell type: even and never divisible by 3. Even powers of 2 and all powers of 10. All known perfect numbers, aside from the first, 6, reside here.
1- and 5- cell type house all prime numbers, aside from 2 and 3. This must be so because all other families are either even, or divisible by three, or both. This is consistent with the well-known congruency of primes with 1 and 5 (mod 6). The 1-type cells form the outer points of the interior hexagram, opposite 4 type cells, visually hinting a connection between the two. Given Mersenne primes (other than 3) land in these cells, and perfect numbers (other than 6) land in the 4 type, the connection seems clearer. It also houses composite numbers, but never composites with 2 or 3 as factors.
Looking at the hexagon, knowing primes land in only the dark blue or light blue triangles, several things become obvious:
- There are two types of prime, aside from 2 and 3.
- Lesser of twin primes are members of the 5 type.
- Greater of twin primes are members of the 1 type.
- Twins are always separated by 0 family members.
- Primes will always have a difference of at least 2, 4 or 6 (or multiples of 6, and multiples of 6 plus 2 or 4.)
- The reasons for ‘sexy primes’ becomes obvious in a six base universe where primes are locked into cells that cycle back every six numbers.
- Two primes differing by 2 or 4 belong to different families.
The above characteristics of primes in the hexagon suggests 0 family numbers split more than twin primes. I speculate these numbers split all primes. That is, all primes have a partner (of the opposite family) equidistant from such a number. For instance, 0 family member 18 splits twin primes 17 and 19, but is also 5 more than 13 and 5 less than 23, and it is also 11 more the 7, and 11 less than 29, etc.