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Re: [PrimeNumbers] Primes not in AP

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  • Jens Kruse Andersen
    ... An AP of primes starting at p cannot have length above p, because p would divide term p+1 which is p + a*p where a is the common difference in the AP. It
    Message 1 of 2 , Mar 26, 2008
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      Robert wrote:
      > are there primes that are not in AP, if AP>2?

      An AP of primes starting at p cannot have length above p,
      because p would divide term p+1 which is p + a*p
      where a is the common difference in the AP.

      It follows from commonly believed conjectures like
      Dickson's conjecture that every prime p is the start
      of infinitely many cases of p primes in AP.
      The largest p with a known case is 17 where the minimal case is
      17 + 341976204789992332560*n for n = 0..16, found by Phil Carmody in 2001:
      http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0111&L=nmbrthry&F=&S=&P=941

      I don't think it has been proved for any prime that it is in
      an AP3 without finding a specific example AP3.

      --
      Jens Kruse Andersen
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