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17874Re: Cubic x^3 + x^2 + x + t prime generators

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  • Mark Underwood
    Mar 29, 2006
      --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>
      wrote:
      >
      > --- Mark Underwood <mark.underwood@...> wrote:
      > > And to top it all off:
      > >
      > > x^8 - x^4 + 1 has no prime factors below 73 (!)
      >
      > Now do you see why PIES has such the incredible density of primes
      that it has?
      > The above is just Phi(24). PIES is looking at Phi(49152) and Phi
      (98304). The
      > super-fruit subprojects can have even higher densities.
      >
      > One puzzle I set the PIES guys was the following:
      > <<<
      > Mathematicians are invited to calculate the relative density of
      primes
      > of the form Phi(24576,715*b^2) compared to arbitrary numbers of the
      > same size.
      > >>>
      > Just finding out what its smallest possible factor is should be an
      eye-opener.
      >

      Phil,
      Something has gone whizzing over my head, and it had to do with Pies
      and Phi's and such. But it looks totally fascninating and I hope to
      learn more along those lines.

      My last result with x^8 - x^4 + 1 producing primes of the form 24n+1
      got me to realize that there might be application to the "Web of
      Ones" thread, and without introducing factors of five into the
      coefficients. And sure enough,

      x^32 - x^24 + x^16 - x^8 + 1

      yields only prime factors ending in one.
      (Except when x=0) Furthermore the prime factors are all of the form
      80n + 1, and there are just 4 different prime factors below 1000:
      241,401,641 and 881.

      Mark
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