Quote:
Originally Posted by henryzz
Finding sets of 6 twins within a range of 81 is easy.
The first is 2595051759329+c with c = 0 2 12 14 30 32 42 44 72 74 78 80
There are several more that are easy to find with polysieve. I am ignoring all other values of c so there could be more than 6 sets of twins assuming a range of 81 is not the minimum.
0 2 6 8 18 20 30 32 36 38 48 50 is possible. I strongly suspect this is minimal for 6 twins. It is only necessary to search 1451 mod 11#=2310 + c for this form.
1256522812841 = 11#*543949269+1451 + c; c= 0 2 6 8 18 20 30 32 36 38 48 50
No proof that this is optimal for 7 but it won't be far off:
11#*491403340492+1451 + c; c = 0 2 6 8 18 20 30 32 36 38 48 50 60 62
I have started a search for 0 2 6 8 18 20 30 32 36 38 48 50 60 62 78 80. I think this will take a while though. Can anyone prove this is optimal?

henryzz, this is great!
 Where are you getting the sets of possible positions?
 How do you derive the primorial and the offset?
Thank you ATH for the very clear explanation of the mods around quads.