Largest Known Cunningham Chains
It follows from Dickson's conjecture and the broader Schinzel's hypothesis H, both widely believed to be true, that for every k there are infinitely many Cunningham chains of length k. There are, however, no known direct methods of generating such chains.
| k | Kind | p1 (starting prime) | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 1 | 1st | 243112609 − 1 | 12978189 | 2008 | GIMPS / Edson Smith |
| 2 | 1st | 183027×2265440 − 1 | 79911 | 2010 | T. Wu |
| 3 | 1st | 914546877×234772 − 1 | 10477 | 2010 | T. Wu |
| 4 | 1st | 119184698×5501# − 1 | 2354 | 2005 | J. Sun |
| 5 | 2nd | 45008010405×2621# + 1 | 1116 | 2010 | D. Broadhurst |
| 6 | 1st | 37488065464×1483# − 1 | 633 | 2010 | D. Augustin |
| 7 | 1st | 162597166369×827# − 1 | 356 | 2010 | D. Augustin |
| 8 | 2nd | 1148424905221×509# + 1 | 224 | 2010 | D. Augustin |
| 9 | 1st | 65728407627×431# − 1 | 185 | 2005 | D. Augustin |
| 10 | 2nd | 1070828503293×239# + 1 | 109 | 2009 | D. Augustin |
| 11 | 2nd | 2×13931865163581×127# + 1 | 63 | 2008 | D. Augustin |
| 12 | 2nd | 13931865163581×127# + 1 | 62 | 2008 | D. Augustin |
| 13 | 1st | 1753286498051×71# − 1 | 39 | 2005 | D. Augustin |
| 14 | 2nd | 335898524600734221050749906451371 | 33 | 2008 | J. K. Andersen |
| 15 | 2nd | 28320350134887132315879689643841 | 32 | 2008 | J. Wroblewski |
| 16 | 2nd | 2368823992523350998418445521 | 28 | 2008 | J. Wroblewski |
| 17 | 2nd | 1302312696655394336638441 | 25 | 2008 | J. Wroblewski |
q# denotes the primorial 2×3×5×7×...×q.
As of 2011, the longest known Cunningham chain of either kind is of length 17. The first known was of the 1st kind starting at 2759832934171386593519, discovered by Jaroslaw Wroblewski in 2008 where he also found some of the 2nd kind.
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