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Re: Prime chains x-->Ax+B

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  • djbroadhurst
    ... For that A does not make sense. If you specify only A, then I may always choose a B such that /no/ proof exists. Hence your maximal length is
    Message 1 of 143 , Dec 1, 2010
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      --- In primenumbers@yahoogroups.com,
      "mikeoakes2" <mikeoakes2@...> wrote:

      > Definition: A "maximal power chain" is a power chain with
      > integers (A,B) for which there is a proof that no chain
      > of greater length exists for that A.

      "For that A" does not make sense. If you specify only A,
      then I may always choose a B such that /no/ proof exists.
      Hence your "maximal" length is unbounded, "for that A".
      However, no worry: we do not need the word "maximal".

      > Definition: L2(A) = (largest prime divisor of (A-1)) - 1.
      > Conjecture: there are only 4 maximal power chains of
      > length >= L2(A)+2.

      So let's make that clear and self contained:

      ! Mike Oakes conjectures that for chains x-->Ax+B, with
      ! A > 2 and B /coprime/ to the /largest/ prime divisor M|A-1,
      ! it is possible to find a chain of M+1 increasing primes
      ! in precisely 4 cases.

      This erases historical confusion and concentrates on what you
      actually studied, with /coprime/ and /largest/ spelled out as
      the central points of a (now) coherent conjecture.

      > $$lots for anyone who can find a 5th one (not:-)

      How many zlotys are you offering :-?

      David
    • djbroadhurst
      ... Suppose that we want to start with a square and get a square. Then we must solve the Diophantine equation y^2 = a*x^2 + b For any pair (a,b), Dario will
      Message 143 of 143 , Jan 7, 2011
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        --- In primenumbers@yahoogroups.com,
        Kevin Acres <research@...> wrote:

        > x=a*x+b either is a square or has a square as a major factor.

        Suppose that we want to start with a square and get a square.
        Then we must solve the Diophantine equation
        y^2 = a*x^2 + b

        For any pair (a,b), Dario will tell us all the solutions:
        http://www.alpertron.com.ar/QUAD.HTM

        David
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