Determine whether or not N is a composite number, where
N = 675*26^{21} + 677*26^{10}  1
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 20060423 13:43:24 