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13175Re: Prime Number Progresions

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  • jbrennen
    Aug 5, 2003
      --- Mark Underwood wrote:

      > Perhaps 2x^2 + 29 generates the longest sequence of consecutive
      > primes of any two term equation.

      If you believe the first Hardy-Littlewood Conjecture (also known
      as the k-tuple Conjecture), there exist arbitrarily long sequences
      of primes from two term equations.

      > 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.
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