From #3 to #52

Testing another sample
Once we had established as much order as possible in the largest primes available (from #9951 to #10000),
the urge to return to the source was indeed very tempting.
The sample chosen is from #3 to #52 in the series of Prime Numbers, at the very beginning.
Here again, each Prime Number is subtracted from the following to establish the Distance (D) between them.

 D2 D4 D2 D4 D2 D4 D6 D2 D6 D4 D2 D4 D6 D6 D2 D6 D4 D2 D6 D4 D6 D8 D4 D2 D4 D2 D4 D14 D4 D6 D2 D10 D2 D6 D6 D4 D6 D6 D2 D10 D2 D4 D2 D12 D12 D4 D2 D4 D6 D2

The Distances behave exactly as predicted with alternation between the right and left columns,
with occasional intrusions from the central column (D6 and, later D12).
The symmetry around F30 (situated in the last D2 of line 1) is quite evident here -
with D6 D4 D2 D4 on each side of it.

 Distance Instances Ratio (%) D2 16 32 D4 15 30 D6 13 26 D10 2 4 D12 2 4 D8 1 2 D14 1 2

The predominance of D2, D4, and D6 is truly remarkable,
especially at the beginning, where we find three sets of D2 D4, nothing but complete Twins,
Prime Numbers at their densest possible state, 1 on 3.

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).

 5 - 7 + 11 - 13 + 17 - 19 + 23 - 29 - 31 + 37 + 41 - 43 + 47 - 53 - 59 - 61 + 67 + 71- 73 + 79 + 83 - 89 - 97 + 101 - 103 + 107 - 109 + 113 - 127 + 131 - 137 - 139 + 149 - 151 + 157 + 163 + 167 - 173 - 179 - 181 + 191 - 193 + 197 - 199 + 211 + 223 + 227 - 229 + 233 - 239 -

Observations
There are approximately the same number of each sign, 26 - and 24 +.
Another sample might offer different proportions.
There will be a 6n Fence immediately above any - Prime and below any + Prime.

Mersenne Primes
As in the case of the Lucas Numbers, it is very tempting to examine and evaluate Mersenne Primes,
built on the algorithm 2p - 1 where p is a Prime number.
If we build them from the beginning, as Mersenne could well have done, -
22 - 1 = 3 (prime #2)
23 - 1 = 7 + (#4)
25 - 1 = 31 + (#11)
27 - 1 = 127 + (#31)
it is quite evident that 24 - 1, 26 - 1, 28 - 1, 29 - 1, 210 - 1, will NOT produce Prime Numbers
211 - 1 = 2047 + (23x89), unfortunately NOT a Prime Number
not good news for the algorithm
213 - 1 = 8191 + (#1028)
Under these circumstances, the Mersenne primes do not deserve
a better grade than that of the Lucas Numbers,
even if the largest prime ever found (in September 2006) was a Mersenne prime.

The Largest Prime
It is difficult to share the infatuation in this never-ending search for larger primes.
What do they prove ? What possible use are they ?
In what other domaine are we always seeking larger numbers ?

Small Is Beautiful
On the contrary, we feel that the greatest beauty and usefulness lies in the smallest primes,
especially the first three (2, 3, and 5) which we see in Applied Maths.

Reversing The Priority
This could, and should, be carried even farther -
(a) that even the first 3 Primes (2, 3, and 5), are not of equal beauty and importance ;
(b) that they are in decreasing order, with the first as monarch over all the others ;
(c) that the importance attached to larger primes is misplaced and harmful ;
(d) that the Mersenne algorithm involved (2p - 1) is essentially binary in concept ;
(e) and that it is high time the coronation of King Prime the First be celebrated as an important event.

Priority Grouping Of Primes
It is interesting to observe in what proportions the three smallest primes
group to form historical numerical references -
The group of 12, the dozen, habitually associated with packaging eggs or donuts,
divides the day into hours, two groups of twelve,
originally from light to darkness, today from mid-night to mid-day.
It is grouped Prime 2 x Prime 2 x Prime 3,
with Prime 2 twice and Prime 3 only once (22x3).
The group of 60, the Babylonian numerical base,
divides the hour into minutes and the minute into seconds,
besides being very present as part of the group of 360 presented below.
It is grouped Prime 2 x Prime 2 x Prime 3 x Prime 5,
with Prime 2 twice, Prime 3 and Prime 5 only once (22x3x5).
The group of 360, the circular numerical base,
divides a rotation into degrees.
It is grouped Prime 2 x Prime 2 x Prime 2 x Prime 3 x Prime 3 x Prime 5,
with Prime 2 three times, Prime 3 twice, and Prime 5 only once (23x32x5).
What more evidence is required to establish the priority of the first three Primes ?
How may solutions are there to the equation n + n = n x n = nn ?
We rest our case.

Conclusion

The Binary System
owes its existence
to the fact that
"2" is the first prime number.