Given a positive integer, check if the number is prime or not. A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of first few prime numbers are {2, 3, 5,

Examples :

Input:  n = 11
Output: true

Input:  n = 15
Output: false

Input:  n = 1
Output: false

School Method
A simple solution is to iterate through all numbers from 2 to n-1 and for every number check if it divides n. If we find any number that divides, we return false.

true
 false

Time complexity of this solution is O(n)

Optimized School Method
We can do following optimizations:

1. Instead of checking till n, we can check till √n because a larger factor of n must be a multiple of smaller factor that has been already checked.
2. The algorithm can be improved further by observing that all primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = ?1, 0, 1, 2, 3, or 4; 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1. (Source: wikipedia)








// A optimized school method based Java // program to check if a number is prime import java.io.*;    class GFG {        static boolean isPrime(int n)     {         // Corner cases         if (n <= 1)             return false;         if (n <= 3)             return true;            // This is checked so that we can skip         // middle five numbers in below loop         if (n % 2 == 0 || n % 3 == 0)             return false;            for (int i = 5; i * i <= n; i = i + 6)             if (n % i == 0 || n % (i + 2) == 0)                 return false;            return true;     }        // Driver Program     public static void main(String args[])     {         if (isPrime(11))             System.out.println(" true");         else             System.out.println(" false");         if (isPrime(15))             System.out.println(" true");         else             System.out.println(" false");     } }

Output:

true
 false

Time complexity of this solution is O(√n)

Main Article : Primality Test | Set 1 (Introduction and School Method)

References:
https://en.wikipedia.org/wiki/Prime_number
http://www.cse.iitk.ac.in/users/manindra/presentations/FLTBasedTests.pdf
https://en.wikipedia.org/wiki/Primality_test

This article is contributed by Ajay. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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