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## 17867Re: Cubic x^3 + x^2 + x + t prime generators

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• 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
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