Prime Numbers

A prime number is a whole number greater than 1, which is only divisible by 1 and itself. First few prime numbers are : 2 3 5 7 11 13 17 19 23 …..

Some interesting fact about Prime numbers

Two is the only even Prime number.
Every prime number can represented in form of 6n+1 or 6n-1 except 2 and 3, where n is natural number.
Two and Three are only two consecutive natural numbers which are prime too.
Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
Wilson Theorem : Wilson’s theorem states that a natural number p > 1 is a prime number if and only if
```
    (p - 1) ! ≡  -1   mod p 
OR  (p - 1) ! ≡  (p-1) mod p
```
Fermat’s Little Theorem: If n is a prime number, then for every a, 1 <= a < n,
```
a^n-1 ≡ 1 (mod n)
 OR 
a^n-1 % n = 1 
```
Prime Number Theorem : The probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
Lemoine’s Conjecture : Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture.

How we check whether a number is Prime or not?

Naive solution.
A naive 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.

C++

// A school method based C++ program to  
// check if a number is prime 
#include <bits/stdc++.h> 

using namespace std; 

// function check whether a number  
// is prime or not 

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 

int main() 
{ 

    isPrime(11) ? cout << " true\n" :  

                  cout << " false\n"; 

    return 0; 
}

Java

// Most optimized Java program to  
// check if a number is prime 

import java.util.*; 

import java.lang.*; 

class GFG { 

    //check for number prime or not 

    static boolean isPrime(int n) { 

    //check if n is a multiple of 2 

    if (n%2==0) return false; 

    //if not, then just check the odds 

    for(int i=3;i<=Math.sqrt(n);i+=2) { 

        if(n%i==0) 

            return false; 

    } 

    return true; 
} 

    // Driver Program 

    public static void main(String[] args)  

    { 

         if(isPrime(19))  

         System.out.println(" true") ; 

         else 

         System.out.println(" false"); 

    } 
} 

// This code is contributed by Ronak Bhensdadia

Python3

# A school method based Python3 program  
# to check if a number is prime 

# function check whether a number  
# is prime or not 

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  

if isPrime(11): 

    print ("true") 

else: 

    print ("false") 

# This code is contributed by Sachin Bisht

// A school method based C# program to  
// check if a number is prime 

using System; 

class GFG 
{ 

    // function check whether a   

    // number is prime or not 

    static 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 Code 

    static void Main()  

    { 

        if(isPrime(11))  

            Console.Write(" true") ; 

        else

            Console.Write(" false"); 

    }  
} 

// This code is contributed by Sam007

PHP

<?php 
// A school method based PHP program to  
// check if a number is prime 

// function check whether a number  
// is prime or not 

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 

if(isPrime(11)) 

    echo("true"); 
else

    echo("false"); 

// This code is contributed by Ajit. 
?>

Output :
Time complexity :O(n)

True

Efficient solutions

Algorithms to find all prime number smaller the N.

More problems related to Prime number

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Prime Numbers

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