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Re: [PrimeNumbers] K-tuples

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  • Jens Kruse Andersen
    ... Tom has added the explanation I was missing earlier. Good. Yes, k*89#+X could be used. Tom does not specify the residue for 79, so k*89#/79+X seems more
    Message 1 of 6 , Jan 29, 2005
      Mike Oakes wrote:

      > > jens.k.a@... writes:
      > >
      > > Conjecturally there exists k such that there are 592 primes from k*592#+N to
      > > k*592#+N+4332.

      > At
      > http://www.opertech.com/primes/w4333.html
      >
      > Tom is using CRT with only the primes up to 89.
      >
      > Does this not mean that the formula could be k*89#+X?

      Tom has added the explanation I was missing earlier. Good.

      Yes, k*89#+X could be used.
      Tom does not specify the residue for 79, so k*89#/79+X seems more natural here.
      It uniquely determines the 592-tuple pattern found by Tom.
      It is only an admissible 592-tuple if there exists k avoiding all factors<=592.
      (Factors>592 clearly have an admissible residue avoiding 592 numbers).
      It can be verified by programming that for all primes p<=592, there are at most
      p-1 different residues (mod p) in Tom's (unwritten) 592-tuple pattern. That
      proves it is an admissible pattern, so the required k's must exist.

      Once you have proved the pattern is admissable, the "formula" could also just be
      "k", i.e. search exhaustively through all numbers.
      My point in using k*592#+N was that we want a 592-tuple. My pfgw directions made
      it trivial for everybody to verify, without programming, that factors<=592 are
      avoided for all k. The actual tuple pattern, not given directly by Tom, can be
      seen as the numbers with no factor in the pfgw output.

      Finding a single k (e.g. the smallest) avoiding factors<=592, by exhaustive
      sieving on k in k*89#/79+X, is infeasible. You can construct large k with CRT but
      then you can just as well use CRT to construct N so you always avoid factors<=592
      in k*592#+N. My N was just one of many with this property.

      --
      Jens Kruse Andersen
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