Newman–Shanks–Williams prime

In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number p which can be written in the form:
${ \Huge S_{2m+1} = \frac{ {(1+\sqrt{2})^{2m+1} + (1-\sqrt{2})^{2m+1} } }{2} }$

Recurrence relation for Newman–Shanks–Williams prime is:

$S_0 = 1$
$S_1 = 1$
$S_n = 2*S_{n-1} + S{n-2}$
The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, ….

Examples:

Input : n = 3
Output : 7

Input : n = 4
Output : 17

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Below is the implementation of finding nth Newman–Shanks–Williams prime:

C++

// CPP Program to find Newman–Shanks–Williams prime 
#include <bits/stdc++.h> 
using namespace std; 
  
// return nth Newman–Shanks–Williams prime 
int nswp(int n) 
{ 
    // Base case 
    if (n == 0 || n == 1) 
        return 1; 
  
    // Recursive step 
    return 2 * nswp(n - 1) + nswp(n - 2); 
} 
  
// Driven Program 
int main() 
{ 
    int n = 3; 
  
    cout << nswp(n) << endl; 
    return 0; 
} 

Java

// Java Program to find  
// Newman-Shanks-Williams prime 
class GFG 
{ 
// return nth Newman-Shanks-Williams 
// prime 
static int nswp(int n) 
{ 
    // Base case 
    if (n == 0 || n == 1) 
        return 1; 
  
    // Recursive step 
    return 2 * nswp(n - 1) + nswp(n - 2); 
} 
  
// Driver code  
public static void main (String[] args) 
{ 
    int n = 3; 
    System.out.println(nswp(n)); 
} 
} 
  
// This code is contributed by Anant Agarwal. 

Python3

# Python3 Program to find Newman–Shanks–Williams prime 
  
# return nth Newman–Shanks–Williams prime 
def nswp(n): 
      
    # Base case 
    if n == 0 or n == 1: 
        return 1
  
    # Recursive step 
    return 2 * nswp(n - 1) + nswp(n - 2) 
  
# Driven Program 
n = 3
print (nswp(n)) 
  
  
# This code is contributed by Shreyanshi Arun. 

C#

// C# Program to find 
// Newman-Shanks-Williams prime 
using System; 
  
class GFG { 
      
    // return nth Newman-Shanks-Williams 
    // prime 
    static int nswp(int n) 
    { 
          
        // Base case 
        if (n == 0 || n == 1) 
            return 1; 
  
        // Recursive step 
        return 2 * nswp(n - 1) + nswp(n - 2); 
    } 
  
    // Driver code 
    public static void Main() 
    { 
        int n = 3; 
          
        Console.WriteLine(nswp(n)); 
    } 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP Program to find  
// Newman–Shanks–Williams prime 
  
// return nth Newman –  
// Shanks – Williams prime 
function nswp($n) 
{ 
      
    // Base case 
    if ($n == 0 || $n == 1) 
        return 1; 
  
    // Recursive step 
    return 2 * nswp($n - 1) +  
               nswp($n - 2); 
} 
  
// Driver Code 
$n = 3; 
echo(nswp($n)); 
  
// This code is contributed by Ajit. 
?> 

Output:

Below is Dynamic Programming solution of finding nth Newman–Shanks–Williams prime:

C++

// CPP Program to find Newman–Shanks–Williams prime 
#include <bits/stdc++.h> 
using namespace std; 
  
// return nth Newman–Shanks–Williams prime 
int nswp(int n) 
{ 
    int dp[n + 1]; 
  
    // Base case 
    dp[0] = dp[1] = 1; 
  
    // Finding nth Newman–Shanks–Williams prime 
    for (int i = 2; i <= n; i++) 
        dp[i] = 2 * dp[i - 1] + dp[i - 2]; 
  
    return dp[n]; 
} 
  
// Driver Program 
int main() 
{ 
    int n = 3; 
  
    cout << nswp(n) << endl; 
    return 0; 
} 

Java

// Java Program for finding 
// Newman-Shanks-Williams prime 
import java.util.*; 
  
class GFG 
{ 
    // return nth Newman_Shanks_Williams prime 
    public static int nswpn(int n) 
    { 
        int dp[] = new int[n + 1]; 
          
        // Base case 
        dp[0] = dp[1] = 1; 
          
        // Finding nth Newman_Shanks_Williams prime 
        for (int i = 2; i <= n; i++) 
          dp[i] = 2 * dp[i - 1] + dp[i - 2]; 
          
        return dp[n]; 
    } 
      
    // Driver Program 
    public static void main (String[] args) { 
          
        int n = 3; 
          
        System.out.println(nswpn(n)); 
    } 
} 
  
/* This code is contributed by Akash Singh */

Python3

# Python3 Program to find  
# Newman–Shanks–Williams prime 
  
# return nth Newman–Shanks 
# –Williams prime 
def nswp(n): 
      
    # Base case 
    dp = [1 for x in range(n + 1)]; 
      
    # Finding nth Newman–Shanks 
    # –Williams prime 
    for i in range(2, n + 1): 
        dp[i] = (2 * dp[i - 1] + 
                     dp[i - 2]); 
    return dp[n]; 
  
# Driver Code 
n = 3; 
print(nswp(n)); 
  
# This code is contributed 
# by mits 

C#

// C# Program to find Newman–Shanks–Williams prime 
  
using System; 
  
class GFG { 
  
    // return nth Newman–Shanks–Williams prime 
    static int nswp(int n) 
    { 
          
        int[] dp = new int[n + 1]; 
  
        // Base case 
        dp[0] = dp[1] = 1; 
  
        // Finding nth Newman–Shanks–Williams prime 
        for (int i = 2; i <= n; i++) 
            dp[i] = 2 * dp[i - 1] + dp[i - 2]; 
  
        return dp[n]; 
    } 
  
    // Driver Program 
    public static void Main() 
    { 
        int n = 3; 
  
        Console.WriteLine(nswp(n)); 
    } 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP Program to find  
// Newman–Shanks–Williams prime 
  
// return nth Newman–Shanks 
// –Williams prime 
function nswp($n) 
{ 
      
    // Base case 
    $dp[0] = $dp[1] = 1; 
  
    // Finding nth Newman–Shanks 
    // –Williams prime 
    for ($i = 2; $i <= $n; $i++) 
        $dp[$i] = 2 * $dp[$i - 1] +  
                      $dp[$i - 2]; 
  
    return $dp[$n]; 
} 
  
// Driver Code 
$n = 3; 
echo(nswp($n)); 
  
// This code is contributed by Ajit. 
?> 

Output:

Below is the code with O(1) space complexity

C++

// C++ code 
#include <iostream> 
using namespace std; 
  
int nswp(int n) 
{ 
      
    if(n == 0 || n == 1)  
    { 
        return 1; 
    } 
      
    // Here we only need to store last 2 values 
    // to find the value of n,  
    // so we will store those 2 values only. 
    int a = 1, b = 1; 
      
    for(int i = 2; i <= n; ++i) 
    { 
        int c = 2 * b + a; 
        a = b; 
        b = c; 
    } 
    return b; 
} 
int main() 
{ 
    int n = 3; 
    cout << nswp(n); 
    return 0; 
} 
  
// This code is contributed by SHUBHAMSINGH10 

Java

//Write Java code here 
class GFG{ 
     static int nswp(int n){ 
           if(n==0 || n==1) return 1; 
           //Here we only need to store last 2 values to find the value of n,  
           //so we will store those 2 values only. 
           int a=1,b=1; 
           for(int i=2;i<=n;++i){ 
               int c=2*b+a; 
               a=b; 
               b=c; 
           } 
           return b; 
     } 
     public static void main(String[] args){ 
          int n=3; 
          System.out.println(nswp(n)); 
     } 
} 

Python3

# Write Python3 code here 
  
def nswp(n): 
    if(n<2): return 1
    a,b=1,1
    for i in range(2,n+1): 
        c=2*b+a 
        a=b 
        b=c 
    return b 
n=3
print(nswp(n)) 

C#

// C# code  
using System; 
  
class GFG  
{ 
    static int nswp(int n) { 
        if (n == 0 || n == 1) 
            return 1; 
              
        // Here we only need to store last 2 values  
        // to find the value of n, 
        // so we will store those 2 values only. 
        int a = 1, b = 1; 
        for (int i = 2; i <= n; ++i) { 
            int c = 2 * b + a; 
            a = b; 
            b = c; 
        } 
        return b; 
    } 
  
    public static void Main(String[] args) 
    { 
        int n = 3; 
        Console.WriteLine(nswp(n)); 
    } 
} 
  
// This code is contributed by PrinciRaj1992 

Output:

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My Personal Notes

Sep	OCT	Nov
	22
2019	2020	2021

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

C++

Java

Python3

C#

PHP

C++

Java

Python3

C#

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