MathNovatory/Pure Maths/Prime Numbers/Distance


Distance Between Primes

Even-numbered Distances
Since the primes are all odd numbers, the distances between them (indicated "D") will all be even numbers. The primes will always be situated more or less symmetrically disposed around a central point of reference.

-(1) D2 +(1)
D4 D6 D8
D10 D12 D14
D16 D18 D20
D22 D24 D26
D28 D30 D32
D34 D36 D38
D40 D42 D44

Lines Of The Table With The Centers Of Symmetry Between the Two Primes
     Using the first few Twin Primes as examples to get started (5 7, 11 13, 17 19) :
if the distance is D2 (from 5 - to 7 +), there will be a 6n Fence (F6) between them ;
if the distance is D4 (from 7 + to 11 -), D6 (from 5 - to 11 - and from 7 + to 13 +), or D8 (from 5 - to 13 +),
     there will be a blitzkrieg (8 9 10) between them ;
if the distance is D10 (from 7 + to 17 -), D12 (from 5 - to 17 - and from 7 + to 19 +), or D14 (from 5 - to 19 +),
      there will be a complete extermination at F12 between them ;
if the distance is D16, D18, or D20, there will be complete exterminations at F12 and F18 ;
if the distance is D22, D24, or D26, there will be complete exterminations at F12, F18, and F24 ;
if the distance is D28, D30, or D32, there will be complete exterminations at F12, F18, F24, and F30 ;
if the distance is D34, D36, or D38, there will be complete exterminations at F12, F18, F24, F30, and F36 ;
if the distance is D40, D42, or D44, there will be complete exterminations at F12, F18, F24, F30, F36, and F42 .

Columns Of The Table With Their Alternation
In the left column, only one distance can be used only once, since they are all from "+" to "-".
In the right column, only one distance (including D2) can be used only once since they are all from "-" to "+".
In the central column, the distances can be repeated or intermingled, the second Prime conserving the same sign.
The right and left columns must thus constantly alternate, spinkled with Distances from the center column.

Testing a sample
The sample chosen is from #9951 to #10000 in the series of Prime Numbers.
     Each Prime Number is subtracted from the following to establish the Distance (D) between them.

D6 D4 D24 D24 D2 D6 D4 D38 D6 D10
D12 D2 D12 D4 D20 D22 D12 D2 D10 D6
D18 D42 D12 D2 D6 D12 D22 D14 D10 D6
D6 D2 D10 D18 D14 D4 D26 D16 D12 D8
D18 D4 D2 D10 D8 D6 D4 D6 D6 D6

The Distances behave exactly as predicted with alternation between the right (- to +) and left (+ to -) columns,
      with occasional intrusions from the central column (D6 D12 D18 D24 D30 D36 D42).


Statistics for this sample
Distance Instances Ratio (%)
D6 11 22
D2 6 12
D4 6 12
D12 6 12
D10 5 10
D18 3 6
D8 2 4
D14 2 4
D22 2 4
D24 2 4
D16 1 2
D20 1 2
D26 1 2
D38 1 2
D42 1 2

Small Distances
It is interesting to note that, even between these relatively large primes, the three smallest distances (D2, D4, and D6) are not only in the first places but occupy 46% (almost half) of the distances, with Twin Primes (D2) occupying 12%, a half-dozen of them.

The Density Of Primes
The density of any span of primes could reliably and accurately be obtained
     by using the inverse of the average of the distances between them.

Establishing - or +
By applying the indications of - and + associated with each Distance size,
     it is possible to establish the sign of each Prime Number, or, in other words, on which side of a 6n Fence it is:
          "-" = -1 (below a 6n Fence), "+" = +1 (above a 6n Fence).

104179 + 104183 - 104207 - 104231 - 104233 + 104239 + 104243 - 104281 + 104287 + 104297 -
104309 - 104311 + 104323 + 104327 - 104347 + 104369 - 104381 - 104383 + 104393 - 104399 +
104417 - 104459 - 104471 - 104473 + 104479 + 104491 + 104513 - 104527 + 104537 - 104543 -
104549 - 104551 + 104561 - 104579 - 104593 + 104597 - 104623 + 104639 - 104651 - 104659 +
104677 + 104681 - 104683 + 104693 - 104701 + 104707 + 104711 - 104717 - 104723 - 104729 -

Observations
There are approximately the same number of each sign, 28 primes are - and 22 are +.
     Another sample might offer different proportions.
The D42 between 104417 and 104459 (third line) signifies seven complete exterminations,
          at F104418, F104424, F104430, F104436, F104442, F104448, and F104454,
     for which we could factor all the neighboring - and + exterminated neighbors.
There will be a 6n Fence immediately above any - Prime and below any + Prime.

Conclusion

Whatever structure there might be in Primes
will be observed in Liberation 2 extermination procedures
rather than in all the existing primes which are only victim survivors.

To Testing Another Sample