1. What is a prime number?

A prime number is a positive integer that can only be divided by 1 and itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. are prime numbers.

Note: 0 and 1 are not prime numbers.

How to check if a positive integer n is a prime number?

1. If n is 0 or 1, it is not a prime number.
2. If n is not divisible by any number from 2 to n-1, then n is a prime number. Otherwise, n is not a prime number.

It is easy to see that only 2 is an even prime number, all other even numbers are not prime because they are divisible by 2. Therefore, instead of checking from 2 to n-1, we can only consider from 2 to n/2.

In addition, it has been proven that: we only need to check if n is divisible by any number from 2 to the square root of n to determine if n is a prime number or not.

2. How to check prime numbers in Java

Using a for loop in Java to check if n is divisible by any number from 2 to n-1 or not?

package primenumber;
import java.util.Scanner;

public class PrimeNumber {
    public static void main(String[] args) {
        int num;
        boolean is_prime = true;
        System.out.print("Enter a positive integer: ");
        try (Scanner scanner = new Scanner(System.in)) {
            num = scanner.nextInt();
        }
        // 0 and 1 are not prime numbers
        if (num == 0 || num == 1) {
            is_prime = false;
        }
        // loop to check if n is prime
        for (int i = 2; i <= num-1; i++) {
            if (num % i == 0) {
                is_prime = false;
                break;
            }
        }
        if (is_prime) {
            System.out.print(num + " is a prime number");
        } else {
            System.out.print(num + " is not a prime number");
        }
    }
}

In this program, we take a number as input from the user and then check whether it is prime or not by dividing it by all possible divisors from 2 to num-1. If the number is divisible by any of these divisors, then it is not a prime number. If none of these divisors can divide the number, then it is a prime number.

We use a boolean variable is_prime to keep track of whether the number is prime or not. We initially assume that the number is prime and set is_prime to true. If we find a divisor that divides the number, then we assigned the value false to is_prime.

At the end of the program, we check the value is_prime to determine whether the number is prime or not.

We can change the loop condition to only run from 2 to n/2.

// loop to check if n is prime
for (int i = 2; i <= num/2; i++) {
    if (num % i == 0) {
        is_prime = false;
        break;
    }
}

You can change the loop condition to only run from 2 to the square root of n (Math.sqrt(n)).

// loop to check if n is prime
for (int i = 2; i <= Math.sqrt(num); i++) {
    if (num % i == 0) {
        is_prime = false;
        break;
    }
}

We should only iterate up to the square root of n to make the loop shorter.

These are three ways how to check prime numbers in Java. The last way is usually the most optimal one for most cases.