As witnessed on Warut Roonguthai's pedagogical http://ksc9.th.com/warut/cunningham.html
, when it comes to record lengths of chains, Cunningham chains of the second kind (p, 2p-1, 4p-3 ...) have the edge over Cunningham chains of the first kind (chains of Sophie Germain primes, p, 2p+1, 4p+3...).
In particular, Tony Forbes in 1997 found CC2s of length 15 and 16 terms, but no known CC1s of equal length were known.
Which is the best reason I know for searching for them!
I wrote some general purpose brute force sieving code[*], and fed it a configuration file which would get it to look for CC1s, and through outrageous luck after 39 hours of searching, I found one of length 15, starting at 113220800675069784839. (I had found none of length 13 or 14 in this time, hence I consider this to be a lucky result)
Due to my sieve configuration, it is quite possible that this is not the smallest example of a CC1_15. Which poses question I hope that I will see settled quite soon.
[*]My thanks go to Paul Jobling for firstly double checking my own algorithms, and providing me with an algorithm that provided the last big speed increase to my code. We are now working together to get a PC version of the code for others to use.
We have to do a bit more testing first though... :-)
Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
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