MathNovatory/Pure Maths/Prime Numbers/3 5

Primes 3 and 5


Liberation 2 with Prime 3

Twin Primes
This is the most structured liberation of them all, removing another 1/6 of the remaining potential primes for a total of 2/3 exterminating composites, leaving 1/3 of the original primes. It is the birth-place of Twin Primes, a very close odd-couple, constantly symmetrically disposed, with a formula of 6n +/- 1.

All Primes Are Twin Primes
The red-blue multiples of 6 (2x3), "F6", "F12", "F18", "F24" (the central column of the illustration on the left) act as Fences delimiting symmetrical disposition of certified primes. The Twin Primes, 5 7, 11 13, 17 19, are to be found on each side of a Fence and their sequence is terminated by the square "25" of green prime 5, which leaves Twin Prime 23 as a survivor of sibling loss. Other following Twin Primes, 29 31, 41 43, 59 61 will be certified later, and 37, 47, 53, 67, also to be certified later, are in reality sibling survivors, and not isolated primes as they seem to be. ALL the following primes will be Twin Primes, neighbors of a 6n Fence, either complete Twin Primes on both sides of the fence, or isolated sibling survivors on one side only.

Complete Extermination
On three occasions between 100 and 200, on the sides of 6n Fences, both Twin Primes are exterminated, leaving gaping prime holes: (1) on the sides of "F120", "119" (7x17) and "121" (11x11), (2) on the sides of "F144", "143" (11x13) and "145" (5x29), (3) on the sides of "F186", "185" (5x37) and "187" (11x17).


Between two adjacent 6n Fences ("F6" and "F12", see above), we find, in the center, a veritable Blitzkrieg of Prime 2 and Prime 3 multiples: "8" (even), "9" (odd multiple of 3), and "10" (even), which will recur between ALL 6n fences. These Blitzkriegs, including the original 2, 3, "4", are on alternate lines which have been removed from the illustration on the left.

Primes Next To 6n Fences
All the following primes must therefore be located next to the 6n Fences.

The Multiples of 3
All the (binary) even multiples of 3 are to be found in the 6n Fences, and all the (binary) odd multiples of 3 are to be found in the very center between Fences, in the middle of a Blitzkrieg.

Liberation 3 with Prime 5

With Fences placed at every 30n (2x3x5), this liberation will remove another 1/15 of the remaining potential Primes for a total of 21/30 exterminating composites, leaving 9/30 (or 3/10) of the original Primes.


Symmetrical Diagonals
We have only one 30n Fence in this liberation because the sequence is terminated by the square "49" of prime 7 (see left). Unable to seek symmetry between 30n Fences because we only have one, we will be obliged to seek it around both sides of the "30" Fence we have, between two 6n Fences, situated at F12 and F48. Twin Primes are present with only 2 exceptions, at "25" (5x5) and at "35" (5x7) which are symmetrically disposed around the central F30 (see above, the three dots represent a Blitzkrieg). The symmetry, within this limited space, is thus perfect, and we will find this diagonal symmetrical disposition everywhere, across each 30n Fence. However, no such new symmetrical dispositions will be discovered, because in a subsequent liberation, the terminating square ("121") comes before the first fence (F210).

A Closer Look at the diagonals
The first diagonal, composed of "25", F30, "35" is very precisely the first line 5, F6, 7 multiplied by 5. As was the case in the preceding liberation, ALL the other diagonals will consist of the corresponding line multiplied by 5.

The Multiples of 5
It might be interesting to identify the positions occupied by the multiples of 5 during a 30n span. The numbers "25", F30, "35" are placed in the first green diagonal (on the left). The numbers "40", "45", "50" are part of three successive Blitzkriegs (not shown on the left). The numbers "55", F60, "65" are placed in the next green diagonal.

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