The Wayback Machine - https://web.archive.org/web/20210412051232/https://www.geeksforgeeks.org/newman-shanks-williams-prime/
Skip to content
Related Articles

Related Articles

Newman–Shanks–Williams prime
  • Difficulty Level : Easy
  • Last Updated : 24 Mar, 2020

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



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:
7

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:
7

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:
7

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
My Personal Notes
arrow_drop_up
Next
last_page
Assembly Line Scheduling | DP-34
Recommended Articles
Page :
Count numbers in a given range having prime and non-prime digits at prime and non-prime positions respectively
15, Jan 21
Quick ways to check for Prime and find next Prime in Java
28, Apr 16
Find coordinates of a prime number in a Prime Spiral
05, Sep 17
Prime numbers after prime P with sum S
27, Feb 18
Sum of each element raised to (prime-1) % prime
24, Dec 18
Print the nearest prime number formed by adding prime numbers to N
26, Jul 18
Absolute Difference between the Sum of Non-Prime numbers and Prime numbers of an Array
02, Nov 18
Absolute difference between the Product of Non-Prime numbers and Prime numbers of an Array
03, Nov 18
Print prime numbers with prime sum of digits in an array
15, Jan 19
Print numbers such that no two consecutive numbers are co-prime and every three consecutive numbers are co-prime
22, Jan 19
Print all numbers whose set of prime factors is a subset of the set of the prime factors of X
22, Apr 19
Sum of prime numbers without odd prime digits
29, Jul 19
Count occurrences of a prime number in the prime factorization of every element from the given range
11, Sep 19
Absolute difference between the XOR of Non-Prime numbers and Prime numbers of an Array
23, Sep 19
Check if a number is Prime, Semi-Prime or Composite for very large numbers
15, Feb 20
Permutation of first N positive integers such that prime numbers are at prime indices
08, Apr 20
Permutation of first N positive integers such that prime numbers are at prime indices | Set 2
29, Apr 20
Count all prime numbers in a given range whose sum of digits is also prime
30, Jun 20
Count prime numbers that can be expressed as sum of consecutive prime numbers
10, Sep 20
Count prime pairs whose difference is also a Prime Number
05, Nov 20
Count numbers in a given range whose count of prime factors is a Prime Number
28, Oct 20
Count distinct prime triplets up to N such that sum of two primes is equal to the third prime
09, Feb 21
Check whether the sum of prime elements of the array is prime or not
26, Nov 18
Maximize difference between sum of prime and non-prime array elements by left shifting of digits minimum number of times
05, Mar 21
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
anuj0503
@anuj0503
Vote for difficulty
Current difficulty :
Easy





Improved By :
Article Tags :
Practice Tags :
Report Issue

We use cookies to ensure you have the best browsing experience on our website. By using our site, you
acknowledge that you have read and understood our
Cookie Policy &
        Privacy Policy

Lightbox