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.
// A school method based JAVA program // to check if a number is prime class GFG { static boolean isPrime( int n) { // Corner case if (n <= 1 ) return false ; // Check from 2 to n-1 for ( int i = 2 ; i < n; i++) if (n % i == 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" ); } } |
true false
Time complexity of this solution is O(n)
Optimized School Method
We can do following optimizations:
- 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.
- 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" ); } } |
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