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17881Re: [PrimeNumbers] Re: Cubic x^3 + x^2 + x + t prime generators

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  • Phil Carmody
    Mar 30, 2006
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      --- Mark Underwood <mark.underwood@...> wrote:
      > > > 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.

      Phi's are cyclotomic polynomials. They are the primitive parts of the family of
      polynomials f(x) = x^n-1. Equivalently they are the minimal polynomial which
      vanishes at each of the primitive n-th roots of unity.
      e.g.
      Phi(2) = x+1, as -1 is the only primitive 2nd root of unity,
      Phi(4) = x^2+1, as +/-i are the only two primitive 4th roots of unity.

      > My last result with x^8 - x^4 + 1 producing primes of the form 24n+1

      That 24 comes from the fact that the above is Phi(24)
      (In Pari/GP, us "polcyclo" to get the n-th cyclotomic polynomial.)

      It's instructive to prove that if p|Phi(n,x) then, with a few exceptions,
      p == 1 (mod n).

      > 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

      Phi(80,x)

      > 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.

      Yup, this is why cyclotomics are rich hunting grounds for primes, as they can't
      have small divisors. My GEFs are typically 2-3 times as dense as Yves' GFNs, as
      I reject even more small factors.

      Phil

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