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19682[PrimeNumbers] Re: Tightened-Lightened Goldbach Conjecture

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  • Jens Kruse Andersen
    Nov 7, 2008
      I had problems understanding the original problem formulation
      so I will try a more formal description.
      Given two natural numbers a < b, find b-a+1 distinct natural
      numbers N_a to N_b such that I +/- N_I is prime for I = a to b.
      In other words, for each integer I from a to b, find a prime
      pair of form I +/- N such that different N is used each time.

      By my hand calculations, if I starts at a=6 then it can at
      most go to b=44. It can do that in four ways, with two possible
      combinations at I=8,10,12, and two options at I=43.
      (I,N_I): (6,1) (7,4) (8,3 or 5) (9,2) (10,7 or 3) (11,6)
      (12,5 or 7) (13,10) (14,9) (15,8) (16,13) (17,14) (18,11)
      (19,12) (20,17) (21,16) (22,15) (23,18) (24,19) (25,22)
      (26,21) (27,20) (28,25) (29,24) (30,23) (31,28) (32,29)
      (33,26) (34,27) (35,32) (36,31) (37,30) (38,35) (39,34)
      (40,33) (41,38) (42,37) (43,36 or 40) (44,39).

      45+/-N is prime for N = 2, 8, 14, 16, 22, 26, 28, 34, 38,
      but they are all taken.

      Bill Krys wrote:
      > ... these are the only prime gaps I can reliably predict
      > where and for hong long they occur.

      The maximal prime gaps at
      http://hjem.get2net.dk/jka/math/primegaps/maximal.htm
      can be used to get an upper limit for how large b can be
      for a given value of a, based on Mark's argument.
      The actual highest value of b may turn out to be lower than
      the limit given in this way.

      --
      Jens Kruse Andersen
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