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.
Statistics for sample 2
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 2^{p}  1 where p is a Prime number. If we build them from the beginning, as Mersenne could well have done,  2^{2}  1 = 3 (prime #2) 2^{3}  1 = 7 + (#4) 2^{5}  1 = 31 + (#11) 2^{7}  1 = 127 + (#31) it is quite evident that 2^{4}  1, 2^{6}  1, 2^{8}  1, 2^{9}  1, 2^{10}  1, will NOT produce Prime Numbers 2^{11}  1 = 2047 + (23x89), unfortunately NOT a Prime Number not good news for the algorithm 2^{13}  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 neverending 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 (2^{p}  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 midnight to midday. It is grouped Prime 2 x Prime 2 x Prime 3, with Prime 2 twice and Prime 3 only once (2^{2}x3). 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 (2^{2}x3x5). 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 (2^{3}x3^{2}x5). 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 = n^{n} ? We rest our case.
Conclusion
The Binary System owes its existence to the fact that "2" is the first prime number.
