Sophie Germain Prime

Write a program to print all sophie germain number less than n. A prime number p is called a sophie prime number if 2p+1 is also a prime number. The number 2p+1 is called a safe prime. For example 11 is a prime number and 11*2 + 1 = 23 is also a prime number so, 11 is sophie germain prime number . The first few Sophie German prime numbers are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179 ..

Examples:

Input : 25
Output : 2 3 5 11 23

Here is the program to print sophie germain number below n .
The solution of this is simple . To obtain all the sophie numbers below n we will make a loop till n and for each number in the loop we can check that whether that number and (2*number + 1), both are prime or not and for checking this we have used Sieve of Erastothenes method.

Below is the implementation of this approach.

C++

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// CPP program to print all sophie german
// prime number till n.
#include <bits/stdc++.h>
using namespace std;
  
// function to detect prime number
// here we have used sieve method
// https://www.geeksforgeeks.org/sieve-of-eratosthenes/
// to detect prime number
bool sieve(int n, bool prime[])
{
    for (int p = 2; p * p <= n; p++) {
  
        // If prime[p] is not changed, then 
        // it is a prime
        if (prime[p] == true) {
  
            // Update all multiples of p
            for (int i = p * 2; i <= n; i += p)
                prime[i] = false;
        }
    }
}
  
void printSophieGermanNumber(int n)
{
    // We have made array till 2*n +1
    // so that we can check prime number
    // till that and conclude about sophie
    // german prime .
    bool prime[2 * n + 1];
    memset(prime, true, sizeof(prime));
    sieve(2 * n + 1, prime);
  
    for (int i = 2; i <= n; ++i) {
  
        // checking every i whether it is
        // sophie german prime or not.
        if (prime[i] && prime[2 * i + 1]) 
            cout << i << " ";        
    }
}
  
int main()
{
    int n = 25;
    printSophieGermanNumber(n);
    return 0;
}

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Java














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// Java program to print all  


// sophie german prime number till n. 


import java.io.*; 


import java.util.*; 


      


class GFG { 


      


    // function to detect prime number 


    // here we have used sieve method 


    // https://www.geeksforgeeks.org/sieve-of-eratosthenes/ 


    // to detect prime number 


    static void sieve(int n, boolean prime[]) 


    { 


        for (int p = 2; p * p <= n; p++) { 


      


            // If prime[p] is not changed, then  


            // it is a prime 


            if (prime[p] == true) { 


      


                // Update all multiples of p 


                for (int i = p * 2; i < n; i += p) 


                    prime[i] = false; 


            } 


        } 


    } 


      


    static void printSophieGermanNumber(int n) 


    { 


        // We have made array till 2*n +1 


        // so that we can check prime number 


        // till that and conclude about sophie 


        // german prime . 


        boolean prime[]=new boolean[2 * n + 1]; 


        Arrays.fill(prime,true); 


        sieve(2 * n + 1, prime); 


      


        for (int i = 2; i < n; ++i) { 


      


            // checking every i whether it is 


            // sophie german prime or not. 


            if (prime[i] && prime[2 * i + 1])  


                System.out.print( i + " ");      


        } 


    } 


      


    public static void main(String args[]) 


    { 


        int n = 25; 


        printSophieGermanNumber(n); 


    } 


} 


  


// This code is contributed  


// by Nikita Tiwari. 



















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Python3














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# Python 3 program to print all sophie  


# german prime number till n. 


  


# Function to detect prime number 


# here we have used sieve method 


# https://www.geeksforgeeks.org/sieve-of-eratosthenes/ 


# to detect prime number 


def sieve(n, prime) : 


    p = 2


    while( p * p <= n ): 


        # If prime[p] is not changed,   


        # then it is a prime 


        if (prime[p] == True) : 


              


            # Update all multiples of p 


            for i in range(p * 2, n, p) : 


                prime[i] = False


                  


        p += 1


          


                  


def printSophieGermanNumber(n) : 


    # We have made array till 2*n +1 


    # so that we can check prime number 


    # till that and conclude about sophie 


    # german prime . 


    prime = [True]*(2 * n + 1) 


      


    sieve(2 * n + 1, prime) 


  


    for i in range(2, n + 1) : 


          


        # checking every i whether it is 


        # sophie german prime or not. 


        if (prime[i] and prime[2 * i + 1]) : 


            print( i , end = " ") 


              


  


# Driver Code 


n = 25


printSophieGermanNumber(n) 


  


  


# This code is contributed by Nikita Tiwari. 



















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C#














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// C# program to print  


// all sophie german  


// prime number till n. 


using System; 


      


class GFG 


{ 


      


    // function to detect prime  


    // number here we have used  


    // sieve method  


    // https://www.geeksforgeeks.org/sieve-of-eratosthenes/ 


    // to detect prime number 


    static void sieve(int n, 


                      bool []prime) 


    { 


        for (int p = 2;  


                 p * p <= n; p++)  


        { 


      


            // If prime[p] is  


            // not changed, then  


            // it is a prime 


            if (prime[p] == true)  


            { 


      


                // Update all multiples of p 


                for (int i = p * 2;  


                         i < n; i += p) 


                    prime[i] = false; 


            } 


        } 


    } 


      


    static void printSophieGermanNumber(int n) 


    { 


        // We have made array till  


        // 2*n +1 so that we can  


        // check prime number till  


        // that and conclude about  


        // sophie german prime . 


        bool []prime = new bool[2 * n + 1]; 


        for (int i = 0;  


                 i < prime.Length; i++) 


        { 


            prime[i] = true; 


        } 


        sieve(2 * n + 1, prime); 


      


        for (int i = 2; i < n; ++i)  


        { 


      


            // checking every i whether  


            // it is sophie german prime 


            // or not. 


            if (prime[i] && prime[2 * i + 1])  


                Console.Write( i + " ");      


        } 


    }      


      


    // Driver code 


    static void Main() 


    { 


        int n = 25; 


        printSophieGermanNumber(n); 


    } 


}  


  


// This code is contributed by  


// Manish Shaw(manishshaw1) 



















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<?php 


// PHP program to print  


// all sophie german 


// prime number till n. 


  


// function to detect prime 


// number here we have used  


// sieve method 


// https://www.geeksforgeeks.org/sieve-of-eratosthenes/ 


// to detect prime number 


function sieve($n, &$prime) 


{ 


    for ($p = 2;  


         $p * $p <= $n; $p++)  


    { 


  


        // If prime[p] is  


        // not changed, then  


        // it is a prime 


        if ($prime[$p] == true)  


        { 


  


            // Update all  


            // multiples of p 


            for ($i = $p * 2; 


                 $i <= $n; $i += $p) 


                $prime[$i] = false; 


        } 


    } 


} 


  


function printSophieGermanNumber($n) 


{ 


    // We have made array till 


    // 2*n +1 so that we can  


    // check prime number till 


    // that and conclude about  


    // sophie german prime . 


    $prime = array(); 


    for($i = 0;  


        $i < (2 * $n + 1); $i++) 


        $prime[$i] = true; 


  


    sieve(2 * $n + 1, $prime); 


  


    for ($i = 2; $i <= $n; ++$i) 


    { 


  


        // checking every i  


        // whether it is sophie 


        // german prime or not. 


        if ($prime[$i] && 


            $prime[2 * $i + 1])  


            echo ($i . " ");      


    } 


} 


  


// Driver code 


$n = 25; 


printSophieGermanNumber($n); 


  


// This code is contributed by  


// Manish Shaw(manishshaw1) 


?> 



















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Output :


2 3 5 11 23



Application of Sophie Prime Numbers :



1. It is used in cryptography as safe primes become the factors of a secret key in RSA cryptosystem.

2. In the first version of AKS Primality Test, it is used to lower the worst case complexity .

3. It is used in the generation of Pseudo Random Number .




          
          
          
          

            

    

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