--- 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