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Re: [PrimeNumbers] FW primes in arithmetic progressions

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  • Jens Kruse Andersen
    ... http://primes.utm.edu/glossary/page.php?sort=DicksonsConjecture says: Dickson conjectured in 1904 that given a family of linear functions with integer
    Message 1 of 4 , Jan 28, 2006
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      Adam wrote:
      > Off the cuff, I don't see how Dickson's cinjecture solves the
      > problem as posed.

      http://primes.utm.edu/glossary/page.php?sort=DicksonsConjecture says:
      Dickson conjectured in 1904 that given a family of linear functions
      with integer coefficients a_i >= 1 and b_i:
      a_1*n + b_1, a_2*n + b_2, ... a_k*n + b_k,
      then there are infinitely many integers n > 0 for which these are
      simultaneously prime unless they "obviously cannot be" (that is,
      unless there is a prime p which divides the product of these for all n).

      Let p be a given prime. Let k=p-1. Let b_i=p and a_i=i for i=1...k.
      Then the conjecture says there exists n > 0 with p-1 simultaneous primes:
      1n+p, 2n+p, ..., (p-1)n+p
      Together with p, this gives the wanted progression of length p starting at p.

      > Your second link shows an ap of length 23
      >
      > 23 56211383760397 + 44546738095860┬Ěn 0..22 16 2004
      > Markus Frind, Paul Jobling, Paul Underwood
      >
      > but that doesn't start at 23!

      That's why I wrote:
      > The minimal progression for p<=17 is listed in
      > http://hjem.get2net.dk/jka/math/aprecords.htm#minimalstart
      >
      > Existence has not been proved for any larger p.

      23>17.
      The AP23 at the link is red, meaning it is the best known but
      (probably) not minimal.

      --
      Jens Kruse Andersen
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