Determine whether or not N is a composite number, where

NOTE:
A prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The numbers 0 and 1 are considered to be neither prime nor composite.

Using e.g.'s transformation
26^23 - 26^21 + 26^12 + 26^10 - 1
I'm trying to factor this.  Why not rewrite it as a polynomial first:
f(x)= x^23 - x^21 + x^12 + x^10 - 1
Now to check if it factors (as a polynomial) for any value of x.

The rational zero theorem says it can only have rational zeroes for x={1,-1}, but f(1)=f(-1)=1  So it is a prime polynomial over the rational numbers. 

So if it can't be factored for any value of x it can't be factored when x=26

Is this valid reasoning?

(By the way f(x) has three real zeroes: -1.1729, -.9361, .94147)

I just realized what was bothering me.  If this was valid reasoning it would imply a polynomial that only generates primes, which is known not to exist.

Posted by Jer on 2006-04-23 13:43:24