## Cubic x^3 + x^2 + x + t prime generators

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• Date: Tue, 28 Mar 2006 03:59:23 -0000 From: Mark Underwood Subject: Re: prime generating quadratic conjecture Hello Patrick, My
Message 1 of 8 , Mar 28, 2006
Date: Tue, 28 Mar 2006 03:59:23 -0000
From: "Mark Underwood" <mark.underwood@...>
Subject: Re: prime generating quadratic conjecture

Hello Patrick,
My guess is that many polynomials of degree 3 and up will be found
which yield over 40 positive and distinct prime numbers in a row. But
if the coefficients are made to equal one, like x^3 + x^2 + x + t, I'm
guessing that none will beat Euler's x^2 + x + 41.

Mark

From Kermit < kermit@... >

In order that x^3 + x^2 + x + t not have any positive prime factors < 43,

t must be

2 mod 3

and

3 or 4 mod 5

and

2 or 5 mod 7

and

2 or 3 or 7 or 9 mod 11

and

3 or 5 or 9 or 11 mod 13

and

2 or 5 or 7 or 8 or 10 or 13 mod 17

and

3 or 4 or 7 or 9 or 12 or 13 mod 19

and

3 or 4 or 5 or 10 or 11 or 12 or 16 or 22 mod 23

and

2 or 5 or 8 or 9 or 13 or 14 or 17 or 20 or 24 or 27 mod 29

and

2 or 10 or 13 or 14 or 16 or 19 or 24 or 27 or 29 or 30 mod 31

and

2 or 4 or 7 or 10 or 14 or 18 or 19 or 24 or 25 or 29 or 33 or 36 mod 37

and

3 or 4 or 7 or 10 or 12 or 15 or 16 or 24 or 25 or 28 or 30 or 33 or 36 or
37 mod 41
• ...
Message 2 of 8 , Mar 28, 2006
--- In primenumbers@yahoogroups.com, "Kermit Rose" <kermit@...> wrote:
>
>
> In order that x^3 + x^2 + x + t not have any positive prime factors
< 43,
>
> t must be
>
> 2 mod 3
>
> and
>
> 3 or 4 mod 5
>
> and
>
> 2 or 5 mod 7
>
> and
>
> 2 or 3 or 7 or 9 mod 11
>
> and

> (snip)

Kermit, what is funny is that I overlooked the obvious:
x^3+x^2+x+k contains a factor of two for every other x! Brought down
by the lowly factor of two.

However here are some things to chew on:

As we know, x^2 + x + 41 contains no prime factors below 41.

But who knew that -x^10 + x^2 + 43 contains no prime factors below
43? And in a strange twist, x^10 - x^2 + 43 also contains no prime
factors below 43, even though their factor set appears to be
different. x^10 - x^2 + 43 has as its lowest factors 43,73,89,107 and
113.

Even more fun, x^8 + x^4 + 59 has no prime factors below 59.

And this is interesting:

x^8 - x^6 - x^4 -x^2 + 1 has no prime factors under 43.

And to top it all off:

x^8 - x^4 + 1 has no prime factors below 73 (!)

Even more incredible is that its lowest five prime factors are
73,97,193,241,337. This is such a glaring scarcity of low prime
factors. I can see that the primes must be of the form 6n+1, but
these appear to go beyond that, to the form 24n+1.

Alot to investigate here, such as the existence of similar forms
which are even poorer in prime factors.

Mark
• ... 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
Message 3 of 8 , Mar 28, 2006
--- 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

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• ... that it has? ... (98304). The ... primes ... eye-opener. ... Phil, Something has gone whizzing over my head, and it had to do with Pies and Phi s and such.
Message 4 of 8 , 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
• ... 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
Message 5 of 8 , Mar 30, 2006
--- 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

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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• ... I don t know if this helps, but the following series will take some beating, albeit it is not cubic: k^36673176307-k^36673176306 from k =1 to infinity has
Message 6 of 8 , Mar 30, 2006
--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- Mark Underwood <mark.underwood@...> wrote:

> > > <<<
> > > 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.
> > >
> >

I don't know if this helps, but the following series will take some
beating, albeit it is not cubic:

k^36673176307-k^36673176306 from k =1 to infinity has no factors less
than 22664022957727 !!!!!

The problem with these polys is that, whilst the factors are of a
certain form and are a quite often a minuscule subset of the primes,
their relative frequence of occurrence is quite high therefore the
density is less attractive than at first glance.

Maybe it is different for cubics but their derivation, described by
Phil in his reply to you, are similar for the poly k^n-k^(n-1)

Regards

Robert Smith
• ... the family of ... polynomial which ... Thank you Phil, Jose and Jack for shedding some light on the theory and showing some pretty brilliant methodologies
Message 7 of 8 , Mar 30, 2006
--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...>
>wrote:>
> 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.

Thank you Phil, Jose and Jack for shedding some light on the theory
and showing some pretty brilliant methodologies to boot. And showing
the capabilities of GP Pari. I have alot to mull over.

Robert, k^n-k^(n-1) has of course factors of k so I assume you did a
typo somewhere?

Mark
• ... Oops, how did I get this muddled? Too many beers when I wrote the response most probably. I meant to say: k^p-(k-1)^p, where p is prime Regards Robert
Message 8 of 8 , Apr 1, 2006
<mark.underwood@...> wrote:
>

>
> Robert, k^n-k^(n-1) has of course factors of k so I assume you did a
> typo somewhere?
>
> Mark
>

Oops, how did I get this muddled? Too many beers when I wrote the
response most probably. I meant to say:

k^p-(k-1)^p, where p is prime

Regards

Robert Smith
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