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 1That 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