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Old 2019-04-30, 19:36   #7
Just call me Henry
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Sep 2007
Cambridge (GMT/BST)

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Originally Posted by robert44444uk View Post
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.
I created the sets of possible positions by hand. After 0 2 and 6 8, 3 can't divide 0 mod 6 or 2 mod 6 and it is clear that 4 mod 6 is divisible by 3 so they can be excluded. After choosing the next twin pair 18 20, 4 mod 10 was the only even number mod 10 that hadn't been seen already so 4 mod 10 must be divisible by 5. I made two lists of numbers excluded by these two criteria, used them to work out a list of possible numbers and then excluded cases that weren't twins. It helped that I could try a case by putting it into polysieve to see if it was possible. I am fairly confident I have optimal solutions but there was a couple of places where I could have chosen other twins.

In this case the offset was calculated by polysieve which uses a wheel sieve. It basically crossed out all offsets <= 11# where one number in the tuple would be divisible by 2 then 3 etc upto 11. In this case there was only one remaining. Polysieve actually continues doing this in order to gain more speed. There are only two possible offsets modulo 13# which leaves 2/13 of the sieve size if I ignored that. I am currently testing 45900 possible offsets for the 8 twins case calculated using primes upto 31 in a modified version of polysieve.

I can go into more detail if necessary.

Last fiddled with by henryzz on 2019-04-30 at 19:37
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