--- 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