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

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

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C#

PHP

C++

Java

Python3

C#

PHP

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