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13182Re: Prime Number Progressions

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  • Robert
    Aug 5, 2003
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      --- In primenumbers@yahoogroups.com, "Robert" <100620.2351@c...>
      wrote:
      >
      > > >
      > > > > But for expressions of the form x^2 + x + p there is no need
      to
      > > > > look for a longer one since it has been shown that p = 41
      > > > > generates the longest one.
      > > >
      > > > Again, the same conjecture implies that arbitrarily long
      > sequences
      > > > of primes exist of the form x^2+x+p.
      > > >
      > > > What has been shown is that p=41 is the largest prime such that
      > > > x^2+x+p is prime for all x, 0 <= x <= p-2.
      > > >
      > > > It has not been shown that x^2+x+p is never prime for 0 <= x
      <=
      > 40.
      >
      > Surely all one has to do is find a c in the equation x^2+x+c for
      > which the following conditions apply:
      >
      > 2+c not divisible by 2, 3, 5
      > and, 2+c meets all of:
      >
      > 1,5,6 mod 7
      > 3,6,8,9,10 mod 11
      > 1,3,4,5,6,7,10 mod 13
      > 1,3,4,5,8,9,10,12 mod 17
      > 4,5,8,11,12,13,14,16,18 mod 19
      > 1,3,9,10,11,12,14,16,17,20,21 mod 23
      > 3,5,6,7,9,10,12,13,14,16,21,22,26,27 mod 29
      > 4,7,11,12,14,15,17,18,19,20,24,26,28,29,30 mod 31
      > 1,6,7,8,10,11,12,13,15,16,17,22,24,25,28,32,35,36 mod 37
      > 3,4,5,6,7,9,11,14,16,18,19,20,21,22,26,27,30,36,39,40 mod 41
      > 1,5,6,8,10,11,14,17,19,22,23,24,26,27,28,29,30,34,36,37,38 mod 43
      >
      > c=41 is the first number to reach all of the conditions except the
      > last, being 1mod7, 10mod11, 4mod13....but 0mod43
      >
      > Regards
      >
      > Robert Smith
      >
      > PS I may have gotten some of the register above incorrect, but
      > someone will spot an error if I have made one. Thats what I like
      > about you Primenumbers group.
      >
      > PPS 2+c = x^2+x+c with x=1
      >
      > PPPS this is the same logic as used in the determination of Payam
      > numbers
      #
      Oops, I forgot to mention that the value of c, which contributes to
      a run of >41 primes, must also clear similar mod n hurdles up to
      sqrt c
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