17867Re: Cubic x^3 + x^2 + x + t prime generators
- Mar 28, 2006--- In firstname.lastname@example.org, "Kermit Rose" <kermit@...> wrote:
> In order that x^3 + x^2 + x + t not have any positive prime factors
>Kermit, what is funny is that I overlooked the obvious:
> t must be
> 2 mod 3
> 3 or 4 mod 5
> 2 or 5 mod 7
> 2 or 3 or 7 or 9 mod 11
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
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.
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