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++
// 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 // 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; } |
Java
// 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 // 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. |
Python3
# Python 3 program to print all sophie # german prime number till n. # Function to detect prime number # here we have used sieve method # 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 = 25printSophieGermanNumber(n) # This code is contributed by Nikita Tiwari. |
C#
// 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 // 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) |
PHP
<?php // PHP program to print // all sophie german // prime number till n. // function to detect prime // number here we have used // sieve method // 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) ?> |
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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