about how to determine a prime number

(1) if divided by 2 or 3, then no;

(2) we only have to go through prime factors; because a composite can always be divided into primes.

(3) since 2 is the smallest prime, for number N, we only have to go through till N/2 max., because if one number is not a prime, the other factor must be no less than 2;

(4) consider N=n*m. If n<sqrt( N ), then it’s a must that m>sqrt( N ). So we only have to go through till sqrt( N )+1 max., because if there’s not a factor with in [2, sqrt(N)+1], there wouldn’t be one above;

(5) other than 2 and 3, prime numbers trend to have a format of (6n +/- 1), but not vise versa.

Now, I haven’t seen a strick mathematical prove on that theory, but someone has run a promgram certifying that at least the first 1 million prime numbers fit in that conclusion.

So if the number is not insanely big, it’s true.

That being say, if we divide a number by (6n +/- 1), it would include many non-prime dividers of course, but we are able to cover all prime factors, too.

Followed is one example:

		l = (int) Math.sqrt (n) + 1;
		for (i=6; i<=l; i+=6) {
			if (n % (i + 1) == 0) return false;
			if (n % (i - 1) == 0) return false;
		}

		// must be prime

(6) seive of Eratosthenes

https://zh.wikipedia.org/zh-hans/%E5%9F%83%E6%8B%89%E6%89%98%E6%96%AF%E7%89%B9%E5%B0%BC%E7%AD%9B%E6%B3%95

The running time for this algorithm is: O = nlog(logn).A pseudo code as followed:

Input: an integer n > 1
 
Let A be an array of Boolean values, indexed by integers 2 to n,
initially all set to true.
 
 for i = 2, 3, 4, ..., not exceeding √n:
  if A[i] is true:
    for j = i2, i2+i, i2+2i, i2+3i, ..., not exceeding n :
      A[j] := false
 
Output: all i such that A[i] is true.

Use seive of Eratosthenes would greatly improve the screening speed. Followed is one example:

	public static void main (String args[]) {
		int	i, j, l;
		A = new boolean[N+1];

		// do a sieve of Eratosthenes

		for (i=0; i<=N; i++) A[i] = true;
		l = (int) Math.sqrt (N);

		// for each number i from 2 to square root of N...

		for (i=2; i<=l; i++) 

			// ...mark off all the multiples of i

			for (j=i*i; j<=N; j+=i) A[j] = false;

		// count whatever is left; these are all the primes

		for (i=2,j=0; i<=N; i++) if (A[i]) j++;
		System.out.println (j);
	}