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Re: [PrimeNumbers] Web--of--Ones

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  • Ignacio Larrosa Cañestro
    Wednesday, March 15, 2006 8:11 AM [GMT+1=CET], ... Thanks to Phil and Jose, I was slightly obtuse ... But, someone could tell us how to prove that it must be p
    Message 1 of 23 , Mar 19, 2006
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      Wednesday, March 15, 2006 8:11 AM [GMT+1=CET],
      Phil Carmody <thefatphil@...> escribió:

      > --- aldrich617 <aldrich617@...> wrote:
      >> I am searching for a way to prove the following Theorem:
      >>
      >> For any integer 'B', the value 'A' of the equation
      >> A = 5B^4 -10B^3 + 20B^2 -15B +11 will have as factors only
      >> integers that end in a one, excluding all others.
      >>
      >> This seems to be true at least up to 10^18. I think that proving
      >> it could give us new insights into primality testing,
      >> and factoring. Moreover there are similar equations, vast in
      >> number, apparently with the same property, that could then probably
      >> be verified to be similar threads of pure one. Together these
      >> would form an infinite interconnecting web.
      >
      > Can you tell us how you discovered that polynomial?
      > The discriminant is very smooth, being 5^3*11^2, and I suspect that
      > that's an essential ingredient to a proof.
      >
      > It seems that if p== +/-1 mod 10, then your polynomial splits at
      > least into 2 quadratics, and if p== +1 mod 10, then at least one of
      > those quadratics splits.
      >
      > This property should be easily explainable, but alas it's late and my
      > brain's on holiday. (I wrote this last night, but forgot to send).

      Thanks to Phil and Jose, I was slightly obtuse ...

      But, someone could tell us how to prove that it must be p = 1 (mod 10)?

      Thanks in advance,

      Ignacio Larrosa Cañestro
      A Coruña (España)
      ilarrosa@...
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