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
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:
- Check first if the given number is a prime number or not. To check for a number to be prime, refer this.
- 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.
- 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)