Jun 2003
Oxford, UK
1,933 Posts

Solved!!!!!
For the last two years I set myself a formidable target, to find a k such that there are 10 or more twins, Cunningham chains of the second kind and Sophie Germains. The challenge is called the triple double, after the equivalent (but easier) basketball feat.
Finally, after a concerted effort during the last week, I can announce the first such k discovered, 13007751179860962525, and I consider myself lucky to find it.
The value have the following 10 Sophie Germain pairs:
[3,4], [4,5], [12,13], [28,29], [90,91], [187,188], [415,416], [487,488], [587,588], [889,890]
14 Cunningham chains of the second kind:
[2,3], [3,4], [4,5], [5,6], [6,7], [26,27], [32,33], [41,42], [68,69], [100,101], [131,132], [154,155], [454,455]
And 12 twins:
3, 4, 5, 7, 12, 17, 26, 29, 33, 42, 69, 155
I have taken n to 10000 without any more pairs of interest.
The k value is a deficient primorial of the form 23#/34 and is a t12 double payam number. I found the value by searching for twins, which are most difficult form of the three, amongst t12 double payams, I checked many millions of t12 values, with a preliminary sieve to only further process those with no factors of less than 257 in the first 6n, and finding those with at least 3 twins at n=6, 7 twins at n=100, and 9 twins at n=500. Of the 185 values with 10 or more twins, only two values gave 10 or more SGs and 4 gave 10 or more CC2s. For this reason I consider it lucky that one value was in both camps.
I would like to give thanks at “Axn1” for his fast double payam number generator, and to “Citrix” for his fast twin counter. Values were rechecked for prp3 using pfgw
Regards
Robert Smith
