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?
