Cunningham Chain - Largest Known Cunningham Chains

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.

Largest known Cunningham chain of length k (as of 8 November 2011 (2011 -11-08))
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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