Hi,
Here's some more data using primorial 210 instead of primorial 30 equations showing distribution of prime last digits:
There are too many permutations to show for all 210 equations that show the last digits of consecutive primes, but here are some highlight results of a subset of the 48 primorial210 equations:
y=210x+b where b = {1 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 121 127 131 137 139 143 149 151 157 163 167 169 173 179 181 187 191 193 197 199 209}
Here are some of the overall set of procurance permutations for the above equations:
primeA, primeA+1, count
97 101 581
37 41 578
17 19 572
113 121 571
73 79 568
29 31 566
31 37 562
193 197 560 ...
151 173 131
19 41 131
187 209 130
127 149 130
113 139 128
169 191 127
121 143 127
67 89 125
31 53 124
101 121 124 ...
59 103 31
149 181 31
101 137 30
61 101 30
53 89 30
97 131 30
167 197 29
181 13 29
131 167 29 ...
53 103 6
197 43 6
97 139 6
97 143 6
19 67 6
143 191 6
23 67 6
89 139 6
19 61 6
31 83 6 ...
83 157 1
11 73 1
23 107 1
173 43 1
169 47 1
131 191 1
23 83 1
83 151 1
103 179 1
67 121 1
181 31 1
That is a huge discrepancy of consecutive last digits occurrences that can be explained with simple statistics, based on how far ahead in number gap the next prime is, ie the top occurring last digit grouping "97 to 101" occurred 581 times, and this is a 4 digit spacing, while the lower occurring last digit groupings, ie "181 to 31" at the bottom, only occurred 1 time. This is a (209-181)+31=59 number gap in the primes, so it makes sense that it occurred less often than the closer spaced primes.
The pairs like "17 to 17" didn't even show up once in the test I did since that is 210 digits apart for consecutive primes within a primorial210 block.
The same thing can be done for larger primorial equation sets, ie primorial 2310 with 480 equations and 480^2 permutation of the consecutive last digit pairings will show a huge discrepancy in the last digits (up to last 4 digits) of consecutive prime numbers.
Here is the C# code excerpt used to do this test:
foreach (int prime in Primes) { foreach (int i in new[] { 1, 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, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163,
167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209 }) { if (((prime - i) % 210) == 0) {
x210plusSORTED.Add(i); } } }
Dictionary pairCounts = new Dictionary(); ChirpletPairCounts chirpletPairCounts = new ChirpletPairCounts();
pairCounts = chirpletPairCounts.GetPairCounts(x210plusSORTED);
var pairCountsSorted = pairCounts.Values.OrderByDescending(x => x);
class ChirpletPairCounts {
public ChirpletPairCounts() {
}
public Dictionary GetPairCounts(IList sequence) { Dictionary pairCounts = new Dictionary(); for (int i = 0; i < sequence.Count - 1; i++) { var Key = Tuple.Create(sequence[i], sequence[i + 1]); if (pairCounts.ContainsKey(Key)) { pairCounts[Key]++; } else { pairCounts.Add(Key, 1); } } return pairCounts; }
}
cheers, Jamie