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Check whether the given number is Wagstaff prime or not

  • Difficulty Level : Easy
  • Last Updated : 26 Apr, 2021

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 a 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 a 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 the above approach:  

C++




// 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;
}

Java




// 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");
        }
    }
}

Python3




# 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")

C#




// 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

PHP




<?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
?>

Javascript




<script>
 
// JavaScript 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 || n % 3 == 0)
            return false;
 
        for (var 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
    function isPowerOfTwo(n)
    {
        return (n != 0 )&& ((n & (n - 1)) == 0);
    }
 
// Driver Program
        
     var n = 43;
 
        // Check if number is prime
        // and of the form ( 2^q +1 )/3
        if (isPrime(n) && (isPowerOfTwo(n * 3 - 1))) {
            document.write("YES");
        }
        else {
            document.write("NO");
        }
 
</script>
Output: 
YES

 

Time Complexity: O(n1/2)

Auxiliary Space: O(1)

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