Primality Test | Set 1 (Introduction and School Method)

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

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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.

Below is the implementation of this method.

C++

// A school method based C++ program to check if a  
// number is prime 
#include <bits/stdc++.h> 
using namespace std; 
  
bool 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 to test above function 
int main() 
{ 
    isPrime(11)?  cout << " true\n": cout << " false\n"; 
    isPrime(15)?  cout << " true\n": cout << " false\n"; 
    return 0; 
} 

Java

// 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"); 
          
    } 
} 
  
// This code is contributed  
// by Nikita Tiwari. 

Python3

# A school method based Python3  
# program to check if a number 
# is prime 
  
def isPrime(n): 
  
    # Corner case 
    if n <= 1: 
        return False
  
    # Check from 2 to n-1 
    for i in range(2, n): 
        if n % i == 0: 
            return False; 
  
    return True
  
# Driver Program to test above function 
print("true") if isPrime(11) else print("false") 
print("true") if isPrime(14) else print("false") 
  
# This code is contributed by Smitha Dinesh Semwal 

C#

// A optimized school method based C# 
// program to check if a number is prime 
using System; 
  
namespace prime 
{ 
    public class GFG 
    {      
        public static bool isprime(int n) 
        { 
            // Corner cases 
            if (n <= 1) return false; 
              
            for (int i = 2; i < n; i++) 
                if (n % i == 0) 
                return false; 
   
            return true; 
        } 
          
        // Driver program 
        public static void Main() 
        { 
            if (isprime(11)) Console.WriteLine("true"); 
            else Console.WriteLine("false"); 
              
            if (isprime(15)) Console.WriteLine("true"); 
            else Console.WriteLine("false"); 
        } 
    } 
} 
  
// This code is contributed by Sam007 

PHP

<?php 
// A school method based PHP  
// program to check if a number  
// is prime 
  
function isPrime($n) 
{ 
    // Corner case 
    if ($n <= 1) return false; 
  
    // Check from 2 to n-1 
    for ($i = 2; $i < $n; $i++) 
        if ($n % $i == 0) 
            return false; 
  
    return true; 
} 
  
// Driver Code 
$tet = isPrime(11) ? " true\n" :  
                     " false\n"; 
echo $tet; 
$tet = isPrime(15) ? " true\n" :  
                     " false\n"; 
echo $tet; 
  
// This code is contributed by m_kit 
?> 

Output :

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)

Below is the implementation of this solution.

C++

// A optimized school method based C++ program to check 
// if a number is prime 
#include <bits/stdc++.h> 
using namespace std; 
  
bool 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 to test above function 
int main() 
{ 
    isPrime(11)?  cout << " true\n": cout << " false\n"; 
    isPrime(15)?  cout << " true\n": cout << " false\n"; 
    return 0; 
} 

Java

// 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"); 
          
    } 
} 
  
/*This code is contributed by Nikita Tiwari.*/

Python3

# A optimized school method based  
# Python3 program to check 
# if a number is prime 
  
  
def isPrime(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 or n % 3 == 0) : 
        return False
  
    i = 5
    while(i * i <= n) : 
        if (n % i == 0 or n % (i + 2) == 0) : 
            return False
        i = i + 6
  
    return True
  
  
# Driver Program  
  
if(isPrime(11)) : 
    print(" true") 
else : 
    print(" false") 
      
if(isPrime(15)) : 
    print(" true") 
else :  
    print(" false") 
      
      
# This code is contributed  
# by Nikita Tiwari. 

C#

// A optimized school method based C#  
// program to check if a number is prime 
using System; 
  
class GFG 
{ 
    public static bool 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 Code 
    public static void Main() 
    { 
            if (isPrime(11)) Console.WriteLine("true"); 
            else Console.WriteLine("false"); 
              
            if (isPrime(15)) Console.WriteLine("true"); 
            else Console.WriteLine("false"); 
    } 
} 
  
// This code is contributed by aj_36 
         

PHP

<?php 
// A optimized school method  
// based PHP program to check 
// if a number is prime 
  
function isPrime($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($i = 5; $i * $i <= $n; $i = $i + 6) 
        if ($n % $i == 0 || $n % ($i + 2) == 0) 
        return false; 
  
    return true; 
} 
  
    // Driver Code 
    if(isPrime(11)) 
        echo " true\n"; 
    else
        echo " false\n"; 
          
    if(isPrime(15)) 
        echo " true\n"; 
    else
        echo " false\n"; 
  
// This code is contributed ajit 
?> 

Output :

true
false

Time complexity of this solution is O(√n)

Primality Test | Set 2 (Fermat 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

My Personal Notes

Improved By : jit_t

Practice Tags :

number-theory

Mathematical

Prime Number