Originally Posted by 10metreh
I don't think you've taken on board what axn meant by this:
The point is that, if n is odd, then gcd(n, n+2N) = gcd(n, N) and so you are just checking if n has any common factors with N. But if N is not prime, then the smallest n that has a common factor with N is the smallest prime factor of N, so in fact when you first find n such that gcd(n, n+2N) > 1, that n is a factor of N. So you're just doing inefficient trial division.
I guess that technically counts as a "primality test" in that it does correctly determine primality or otherwise of N.
But it's easy to come up with useless primality tests. Here's one: if N>4, then N is prime if and only if N does not divide (N1)!. Unlike trial division, it's got only one step, so what could possibly go wrong...?
As for why gcd(n, n+2N) = gcd(n, N), it's time for another basic number theory lesson:
Suppose d divides n and n+2N.
Then d must divide their difference, which is 2N.
But d divides n, which is odd, so d is odd, and thus d must divide N.
Conversely, suppose d divides n and N.
Then clearly d divides n+N+N = n+2N.
So the sets of common factors of (n, n+2N) and (n, N) are exactly the same, and in particular gcd(n, n+2N) = gcd(n, N).
