Prime Numbers - GeeksforGeeks

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 

C#

// 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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C++

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