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++

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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;
}

chevron_right






                        filter_none









Java














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// 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. 



















                        chevron_right







                        filter_none









Python3














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















# 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. 



















                        chevron_right







                        filter_none









C#














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















// 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) 



















                        chevron_right







                        filter_none









PHP














                                    filter_none




                                    edit

                                    close




                                    play_arrow




                                    link

                                    brightness_4

                                    code
                                




















<?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) 


?> 



















                        chevron_right







                        filter_none










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 .




          
          
          
          

            

    

        My Personal Notes
        arrow_drop_up
    

    

        
        

            
            
        

    


Recommended Posts:
Surya Priy
Check out this Author's contributed articles.
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on  the "Improve Article" button below.


Improved By :  manishshaw1