Hi,
There are easily seen harmonic patterns in the prime number last digits distributions that can be see when primorial size number blocks are used to count the primes with given last digits. A primorial 30 number block with size 30, has EulerPhi(30)^2 = 64 permutations of consecutive last digits for any two consecutive prime numbers within the 30 sized number block aligned with zero. There is an oscillatory frequency of EulerPhi(30)=8 cycles per 30 number sized block for each of the 8 different first primes, 3,7,11,13,17,19,23,29 in permutations with all the other primes including itself (64 different possible last digit pairings). This pattern holds for all primorials and the corresponding oscillatory frequency is still EulerPhi(primorial).
Here is some data showing more info on this, the 8 cycle beat frequency is easiest to see in the primorial30 block, subsequent primorials above 30 weren't fully analyzed to see all the permutations so only a portion of the beat frequency period (the portion with higher occurance) is visible.
primeA, primeA+1, count
primorial6:
5,5,16427
5,1,22837
1,5,22837
1,1,16394
primorial30: full list of 64 (8*8) permutations
29,29,221
29,23,363
29,19,519
29,17,774
29,13,1345
29,11,1526
29,7,2153
29,1,2904
23,29,2893
23,23,226
23,19,431
23,17,543
23,13,720
23,11,993
23,7,1824
23,1,2209
19,29,2154
19,23,2887
19,19,261
19,17,431
19,13,517
19,11,957
19,7,1020
19,1,1561
17,29,1551
17,23,2181
17,19,2914
17,17,265
17,13,380
17,11,516
17,7,724
17,1,1278
13,29,1160
13,23,1590
13,19,2163
13,17,2892
13,13,274
13,11,448
13,7,560
13,1,737
11,29,783
11,23,1133
11,19,1551
11,17,2193
11,13,2916
11,11,266
11,7,421
11,1,547
7,29,633
7,23,795
7,19,1138
7,17,1624
7,13,2087
7,11,2945
7,7,260
7,1,330
1,29,410
1,23,665
1,19,811
1,17,1087
1,13,1585
1,11,2159
1,7,2849
1,1,241
primorial210: (only excerpts)
209,73,1
209,67,2
209,61,2
209,59,1
209,53,1
209,47,4
209,43,9
209,41,9
209,37,19
209,31,37
209,29,49
209,23,68
209,19,105
209,17,177
209,13,241
209,11,374
209,1,542
199,209,549
199,71,1
199,67,2
199,61,1
199,59,1
199,47,4
199,43,8
199,41,8
199,37,4
199,31,16
199,29,35
199,23,40
199,19,69
199,17,97
199,13,141
199,11,271
199,1,371
197,209,370
197,199,555
197,53,1
197,47,3
197,43,6
197,41,10
primorial2310:
2309,41,2
2309,37,2
2309,31,5
2309,29,11
2309,23,5
2309,19,13
2309,17,28
2309,13,34
2309,1,66
2297,2309,55
2297,47,1
2297,41,1
2297,37,3
2297,31,4
2297,29,4
2297,23,7
2297,19,4
2297,17,18
2297,13,28
2297,1,37
2293,2309,44
2293,2297,60
2293,29,3
2293,23,5
2293,19,5
2293,17,11
2293,13,17
2293,1,19
2291,2309,23
2291,2297,37
2291,2293,59
2291,37,1
2291,29,2
2291,23,2
primorial30030:
30029,29,2
30029,19,3
30029,17,3
30029,1,9
30013,30029,10
30013,29,1
30013,23,1
30013,19,1
30013,1,2
30011,30029,2
30011,30013,6
30011,17,1
30011,1,4
30007,30029,3
30007,30013,7
30007,30011,5
30007,31,1
30007,1,1
30001,30029,2
30001,30013,1
30001,30011,2
30001,30007,4
30001,19,1
29999,30007,5
29999,30001,4
29999,17,2
29993,30007,6
29993,30001,2
29993,29999,5
29989,30013,1
29989,30011,4
29989,30001,1
29989,29999,2
There is a harmonic sine wave visible in the sorted count values, ie visible easily in the above primorial30, with frequency that corresponds to a given primorial block size. The frequency is just
8 per 30 in the case of primorial 30 as seen above. Each primorial has a beat frequency of EulerPhi(primorial)
ie:
primorial beatFrequency
6 2
30 8
210 48
2310 48
30030 5760
ie in wolfram alpha:
These can explain all the harmonics observed in the prime number distributions I think.
For the primorials 210 and above, I didn't run all the permutations, but the partial full wave of oscillation is visible, and with a bit of imagination the full oscillating sine wave of EulerPhi(primorial) can be seen :D
The whole thing could be displayed with a statistical distribution formula instead of manually calculating the primes for it too, as for higher primorials the gaps get too big between some of the permutations to occur enough to show the sine wave plot without a supercomputer. Thats why the pattern becomes less visible for big primorials, but if they are fully plotted they should have nice sine waves of EulerPhi(primorial)
cheers, Jamie