## prime generating polynomials

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• Mark Underwood wrote: However here are some things to chew on: As we know, x^2 + x + 41 contains no prime factors below 41. But
Message 1 of 2 , Mar 29, 2006
Mark Underwood <mark.underwood@...> wrote:

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

Kermit says:

Examine

x^10 - x^2 + 43

mod 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41

First examine x^10 - x^2

x^10 - x^2 = x^2 [ x^8 - 1 ] = x^2 [ x^4 + 1] [ x^2 + 1 ] [x + 1] [ x
- 1]

x^10 - x^2 is always zero mod 2 since x(x-1) = 0 mod 2

x^10 - x^2 is always zero mod 3 since x(x-1)(x+1) = 0 mod 3

x^10 - x^2 is always zero mod 5 since x(x4-1) = 0 mod 5

Mod 7
x^2 : 0
x-1: 1
x+1: 6
x^10 - x^2 is zero mod 7, for x = 0, or x=1 or x = 6

2^10 - 2^2 mod 7 = 5
3^10 - 3^2 mod 7 = 2
4^10 - 4^2 mod 7 = 2
since 3 + 4 = 7
5^10 - 5^2 = 5
since 2 + 5 = 7.

x^8 + x^4 + 59 mod 2, 3, 5, etc

x^8 + x^4 = x^4 ( x^4 + 1)

is always 0 mod 2
is never = 1 mod 3
is never = 1 or 3 or 4 mod 5
etc

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

(x^4)^2 - x^4 + 1 mod 2 = 1

(x^4)^2 - x^4 + 1 mod 3 = 1

(x^4)^2 - x^4 + 1 mod 5 = 1

(x^4)^2 - x^4 + 1 mod 7 = 1 or 3 or 6

etc

> 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.

x^8 - x^4 + 1 is equal to 1 in mod 2, mod 3, mod 4, mod 5, mod 8, mod 16,

and therefore

x^8 - x^4 + 1 is equal to 1 mod 240

Of course this says nothing about what the lowerest prime factors would be.
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