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

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• 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

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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