Possible Prime Constellation Patterns
Here is a list of all the possible prime constellation patterns, up to 24-tuples. This page provides a list up to 50 along with related mathematical constants.
Note that no constellation of length 24 is known, and the unique ones known for 22 and 23 are trivial.
Riecoin uses 7-tuples starting from the second fork (formerly 6-tuples) for Mainnet and 5-tuples for Testnet.
| Length | Offset from start |
|---|---|
| 1 | 0 |
| 2 | 0, 2 |
| 3 | 0, 2, 6 |
| 3 | 0, 4, 6 |
| 4 | 0, 2, 6, 8 |
| 5 | 0, 2, 6, 8, 12 |
| 5 | 0, 4, 6, 10, 12 |
| 6 | 0, 4, 6, 10, 12, 16 |
| 7 | 0, 2, 6, 8, 12, 18, 20 |
| 7 | 0, 2, 8, 12, 14, 18, 20 |
| 8 | 0, 2, 6, 8, 12, 18, 20, 26 |
| 8 | 0, 2, 6, 12, 14, 20, 24, 26 |
| 8 | 0, 6, 8, 14, 18, 20, 24, 26 |
| 9 | 0, 2, 6, 8, 12, 18, 20, 26, 30 |
| 9 | 0, 2, 6, 12, 14, 20, 24, 26, 30 |
| 9 | 0, 4, 6, 10, 16, 18, 24, 28, 30 |
| 9 | 0, 4, 10, 12, 18, 22, 24, 28, 30 |
| 10 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
| 10 | 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 |
| 11 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
| 11 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 |
| 12 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 |
| 12 | 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 |
| 13 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 |
| 13 | 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 |
| 13 | 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 |
| 13 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48 |
| 13 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 |
| 13 | 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 |
| 14 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 |
| 14 | 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 |
| 15 | 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 |
| 15 | 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 |
| 15 | 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 |
| 15 | 0, 6, 8, 14, 20, 24, 26, 30, 36, 38, 44, 48, 50, 54, 56 |
| 16 | 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 |
| 16 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 |
| 17 | 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 |
| 17 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 |
| 17 | 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 |
| 17 | 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 |
| 18 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70 |
| 18 | 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 |
| 19 | 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 |
| 19 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70, 76 |
| 19 | 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76 |
| 19 | 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 |
| 20 | 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 |
| 20 | 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 |
| 21 | 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 |
| 21 | 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 |
| 22 | 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80, 86, 90 |
| 22 | 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84, 90 |
| 22 | 0, 4, 10, 12, 18, 22, 24, 28, 34, 40, 42, 48, 52, 54, 60, 64, 70, 78, 82, 84, 88, 90 |
| 22 | 0, 6, 8, 14, 18, 20, 24, 30, 36, 38, 44, 48, 50, 56, 60, 66, 74, 78, 80, 84, 86, 90 |
| 23 | 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84, 90, 94 |
| 23 | 0, 4, 10, 12, 18, 22, 24, 28, 34, 40, 42, 48, 52, 54, 60, 64, 70, 78, 82, 84, 88, 90, 94 |
| 24 | 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 60, 66, 70, 72, 76, 82, 84, 90, 94, 96, 100 |
| 24 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70, 76, 84, 88, 94, 96, 100 |
| 24 | 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 48, 54, 58, 60, 66, 70, 76, 84, 88, 90, 94, 96, 100 |
| 24 | 0, 4, 6, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84, 90, 94, 96, 100 |
Here are the same patterns, but with non cumulative offsets (useful in some configuration files and implementations).
| Length | Offset differences |
|---|---|
| 1 | 0 |
| 2 | 0, 2 |
| 3 | 0, 2, 4 |
| 3 | 0, 4, 2 |
| 4 | 0, 2, 4, 2 |
| 5 | 0, 2, 4, 2, 4 |
| 5 | 0, 4, 2, 4, 2 |
| 6 | 0, 4, 2, 4, 2, 4 |
| 7 | 0, 2, 4, 2, 4, 6, 2 |
| 7 | 0, 2, 6, 4, 2, 4, 2 |
| 8 | 0, 2, 4, 2, 4, 6, 2, 6 |
| 8 | 0, 2, 4, 6, 2, 6, 4, 2 |
| 8 | 0, 6, 2, 6, 4, 2, 4, 2 |
| 9 | 0, 2, 4, 2, 4, 6, 2, 6, 4 |
| 9 | 0, 2, 4, 6, 2, 6, 4, 2, 4 |
| 9 | 0, 4, 2, 4, 6, 2, 6, 4, 2 |
| 9 | 0, 4, 6, 2, 6, 4, 2, 4, 2 |
| 10 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2 |
| 10 | 0, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 11 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 11 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 12 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6 |
| 12 | 0, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 13 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6 |
| 13 | 0, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 13 | 0, 2, 10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 13 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2 |
| 13 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2 |
| 13 | 0, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 14 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2 |
| 14 | 0, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 15 | 0, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6 |
| 15 | 0, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2 |
| 15 | 0, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2 |
| 15 | 0, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2 |
| 16 | 0, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 16 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2 |
| 17 | 0, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4 |
| 17 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6 |
| 17 | 0, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2 |
| 17 | 0, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 18 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4 |
| 18 | 0, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 19 | 0, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4 |
| 19 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6 |
| 19 | 0, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4 |
| 19 | 0, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 20 | 0, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2 |
| 20 | 0, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2 |
| 21 | 0, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4 |
| 21 | 0, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2 |
| 22 | 0, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4 |
| 22 | 0, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6 |
| 22 | 0, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2 |
| 22 | 0, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4 |
| 23 | 0, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4 |
| 23 | 0, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4 |
| 24 | 0, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 8, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |
| 24 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 6, 2, 4 |
| 24 | 0, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 8, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4 |
| 24 | 0, 4, 2, 6, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4 |