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.
Read more about this topic: Cunningham Chain
Famous quotes containing the words largest, cunningham and/or chains:
“...I believed passionately that Communists were a race of horned men who divided their time equally between the burning of Nancy Drew books and the devising of a plan of nuclear attack that would land the largest and most lethal bomb squarely upon the third-grade class of Thomas Jefferson School in Morristown, New Jersey.”
—Fran Lebowitz (b. 1950)
“While the hollow oak our palace is,
Our heritage the sea.”
—Allan Cunningham (1784–1842)
“Of all men living [priests] are our greatest enemies. If it were possible, they would extinguish the very light of nature, turn the world into a dungeon, and keep mankind for ever in chains and darkness.”
—George Berkeley (1685–1753)