Check whether the given number is Wagstaff prime or not

Given a positive integer n, the task is to check if it is a Wagstaff prime or not. Print ‘YES’ if the given number is Wagstaff prime otherwise print ‘NO’.

Wagstaff prime: In mathematics, Wagstaff prime is a prime number ‘n’ of the form n = \frac{2^{q} + 1}{3}
where ‘q’ is an odd prime.

First few Wagstaff prime numbers are:

3, 11, 43, 683, 2731, 43691, 174763, 2796203……….

Examples:



Input: 43
Output: Yes
43 can be expressed as - (27 + 1 )/ 3

Input: 31
Output: No
31 can not be expressed in above mentioned form.

Approach:

  1. Check first if the given number is prime number or not. To check for a number to be prime, refer this.
  2. Then check if it can be expressed in the form of (n * 3 – 1) and should be a power of 2. To check for a number to be power of 2, refer this.
  3. If both conditions are true, then the number is a Wagstaff prime number. Hence print “YES”. Otherwise print “NO”

Below is the implementation of above approach:

C++

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// CPP program to check if a number is
// Wagstaff prime or not
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to check if a number is prime or not
bool isPrime(int n)
{
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
  
    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
  
    for (int i = 5; i * i <= n; i = i + 6) {
        if (n % i == 0 || n % (i + 2) == 0) {
            return false;
        }
    }
  
    return true;
}
  
// Utility function to check power of two
bool isPowerOfTwo(int n)
{
    return (n && !(n & (n - 1)));
}
  
// Driver Program
int main()
{
    int n = 43;
  
    // Check if number is prime
    // and of the form (2^q +1 )/ 3
  
    if (isPrime(n) && (isPowerOfTwo(n * 3 - 1))) {
        cout << "YES\n";
    }
    else {
        cout << "NO\n";
    }
  
    return 0;
}

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Java

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// JAVA program to check if a number is
// Wagstaff prime or not
  
class GFG {
  
    // Function to check if a number is prime or not
    static boolean isPrime(int n)
    {
        // Corner cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;
  
        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
            return false;
  
        for (int i = 5; i * i <= n; i = i + 6) {
            if (n % i == 0 || n % (i + 2) == 0) {
                return false;
            }
        }
        return true;
    }
  
    // Utility function to check power of two
    static boolean isPowerOfTwo(int n)
    {
        return n != 0 && ((n & (n - 1)) == 0);
    }
  
    // Driver Program
    public static void main(String[] args)
    {
        int n = 43;
  
        // Check if number is prime
        // and of the form ( 2^q +1 )/3
        if (isPrime(n) && (isPowerOfTwo(n * 3 - 1))) {
            System.out.println("YES");
        }
        else {
            System.out.println("NO");
        }
    }
}

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Python3

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# Python 3 program to check if a number is  
# Wagstaff prime or not 
    
# Utility function to check 
# if a number is prime or not 
def isPrime(n) :  
    # Corner cases  
    if (n <= 1) :  
        return False
    if (n <= 3) :  
        return True
    
    # This is checked so that we can skip  
    # middle five numbers in below loop  
    if (n % 2 == 0 or n % 3 == 0) :  
        return False
    
    i = 5
    while(i * i <= n) :  
        if (n % i == 0 or n % (i + 2) == 0) :  
            return False
        i = i + 6
    
    return True
  
# Utility function to Check
# power of two
  
def isPowerOfTwo(n):
      
    return (n and (not(n & (n - 1)))) 
            
# Driver Code  
n = 43
        
# Check if number is prime  
# and of the form ( 2 ^ q + 1 ) / 3
    
if(isPrime(n) and isPowerOfTwo(n * 3-1)): 
    
    print("YES"
    
else
    
    print("NO"

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C#

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// C# program to check if a number 
// is Wagstaff prime or not
using System;
  
class GFG 
{
  
// Function to check if a
// number is prime or not
static bool isPrime(int n)
{
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
  
    // This is checked so that we 
    // can skip middle five numbers 
    // in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
  
    for (int i = 5; 
             i * i <= n; i = i + 6)
    {
        if (n % i == 0 ||
            n % (i + 2) == 0) 
        {
            return false;
        }
    }
    return true;
}
  
// Utility function to 
// check power of two
static bool isPowerOfTwo(int n)
{
    return n != 0 && ((n & (n - 1)) == 0);
}
  
// Driver Code
public static void Main()
{
    int n = 43;
  
    // Check if number is prime
    // and of the form ( 2^q +1 )/3
    if (isPrime(n) && 
       (isPowerOfTwo(n * 3 - 1)))
    {
        Console.WriteLine("YES");
    }
    else 
    {
        Console.WriteLine("NO");
    }
}
}
  
// This code is contributed
// by inder_verma

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PHP

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<?php
// PHP program to check if a number 
// is Wagstaff prime or not
  
// Function to check if a 
// number is prime or not
function isPrime($n)
{
    // Corner cases
    if ($n <= 1)
        return false;
    if ($n <= 3)
        return true;
  
    // This is checked so that we 
    // can skip middle five numbers
    // in below loop
    if ($n % 2 == 0 or $n % 3 == 0)
        return false;
  
    for ($i = 5; 
         $i * $i <= $n; $i = $i + 6) 
    {
        if ($n % $i == 0 or 
            $n % ($i + 2) == 0) 
        {
            return false;
        }
    }
  
    return true;
}
  
// Utility function to 
// check power of two
function isPowerOfTwo($n)
{
    return ($n && !($n & ($n - 1)));
}
  
// Driver Code
$n = 43;
  
// Check if number is prime
// and of the form (2^q +1 )/ 3
  
if (isPrime($n) && 
   (isPowerOfTwo($n * 3 - 1))) 
{
    echo "YES";
}
else 
{
    echo"NO";
}
  
// This code is contributed
// by Shashank
?>

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Output:

YES

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