Am I crazy, or should we not expect to have a few tens of septuplets already? Perhaps I'm mis-thinking the math -- 1/ln(2^1700) ~= 1/1200 chance of a sextuplet being a septuplet?
You're right, but looking at the admissible patterns for sextuplets vs septuplets:
0 4 6 10 12 16
vs
0 2 6 8 12 18 20
0 2 8 12 14 18 20
it doesn't fit...looks like we're screwed... we won't have septuplets with minimal distance (p ... p+20)
I didn't verify this, but the source is Anthony Forbes
same thing with octuplets:
0 2 6 8 12 18 20 26
0 2 6 12 14 20 24 26
0 6 8 14 18 20 24 26
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craigdomnic@gmail.comI spotted that as well, but it might not be as bad as you think (but not as easy as testing one extra prime).
If you take the first p7 variant and subtract 2 you get the pattern:
-2 0 4 6 10 16 18
Aside from "12" is a very good match for the 6-tuplet pattern (for the second p7 variant you ignore the "4"). In both cases you just need to test p-2 and p+18 for a valid p7 chain. In effect a valid 6-tuplet means you know you have 5 out of 7 valid primes for the 7-tuplet.
A quick look at the others shows similar tricks to "re-use" valid 6-tuplets probably also exist.
Once again check my assumptions....
EDIT: For the 7-tuplet, you can also subtract 4 and get another pretty good subset to use as the basis of a test. Should also increase chance of finding a valid chain.
Regards,
--
bsunau7