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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@...>
      > --- 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
      > 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

      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.

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