# Program to check whether a number is Proth number or not

Given a positive integer N, the task is to check if it is a Proth number. If the given number is a Proth number then print ‘YES’ otherwise print ‘NO’.

Proth Number: In mathematics, a Proth number is a positive integer of the form

n = k * 2n + 1

where k is an odd positive integer and n is a positive integer such that 2n > k .

The first few Proth numbers are –

3, 5, 9, 13, 17, 25, 33, 41, 49, ……

Examples:

Input: 25
Output: YES
Taking k= 3 and n= 3,
25 can be expressed in the form of
(k.2n + 1) as (3.23 + 1)

Input: 73
Output: NO
Taking k=9 and n=3
73 can be expressed in the form of
(k.2n + 1 ) as  (9.23 + 1)
But 23 is less than 9
(it should be greater than k to be Proth Number)

Approach

1. Deduct 1 from the number. This would give a number in the form k*2n, if the given number is a proth number.
2. Now, loop through all odd number starting form k=1 to n/k and check if k can divide n in such a way that ( n/k ) is a power of 2 or not.
3. If found, print ‘YES’
4. If no such value of k is found then Print ‘NO’

Below is the implementation of above idea

## C++

 // CPP program to check Proth number    #include using namespace std;    // Utility function to check power of two bool isPowerOfTwo(int n) {     return (n && !(n & (n - 1))); }    // Function to check if the // Given number is Proth number or not bool isProthNumber(int n) {        int k = 1;     while (k < (n / k)) {            // check if k divides n or not         if (n % k == 0) {                // Check if n/k is power of 2 or not             if (isPowerOfTwo(n / k))                 return true;         }            // update k to next odd number         k = k + 2;     }        // If we reach here means     // there exists no value of K     // Such that k is odd number     // and n/k is a power of 2 greater than k     return false; }    // Driver code int main() {        // Get n     int n = 25;        // Check n for Proth Number     if (isProthNumber(n - 1))         cout << "YES";     else         cout << "NO";        return 0; }

## Java

 // Java program to check for Proth number    class GFG {        // Utility function to check power of two     static boolean isPowerOfTwo(int n)     {         return n != 0 && ((n & (n - 1)) == 0);     }        // Function to check if the     // Given number is Proth number or not     static boolean isProthNumber(int n)     {            int k = 1;         while (k < (n / k)) {                // check if k divides n or not             if (n % k == 0) {                    // Check if n/k is power of 2 or not                 if (isPowerOfTwo(n / k))                     return true;             }                // update k to next odd number             k = k + 2;         }            // If we reach here means         // there exists no value of K         // Such that k is odd number         // and n/k is a power of 2 greater than k         return false;     }        // Driver code     public static void main(String[] args)     {            // Get n         int n = 25;            // Check n for Proth Number         if (isProthNumber(n - 1))             System.out.println("YES");         else             System.out.println("NO");     } }

## Python3

 # Python3 program to check for Proth number            # Utility funtion to Check  # power of two  def isPowerOfTwo(n):               return (n and (not(n & (n - 1))))                 # Function to check if the # Given number is Proth number or not def isProthNumber( n):               k = 1            while(k < (n//k)):                    # check if k divides n or not         if(n % k == 0):                # Check if n / k is power of 2 or not             if(isPowerOfTwo(n//k)):                     return True                        # update k to next odd number         k = k + 2                          # If we reach here means     # there exists no value of K     # Such that k is odd number       # and n / k is a power of 2 greater than k     return False                                          # Driver code    # Get n     int n = 25;    # Check n for Proth Number if(isProthNumber(n-1)):     print("YES"); else:     print("NO");

## C#

 // C# program to check Proth number    using System; class GFG {        // Utility function to check power of two     static bool isPowerOfTwo(int n)     {         return n != 0 && ((n & (n - 1)) == 0);     }        // Function to check if the     // Given number is Proth number or not     static bool isProthNumber(int n)     {            int k = 1;         while (k < (n / k)) {                // check if k divides n or not             if (n % k == 0) {                    // Check if n/k is power of 2 or not                 if (isPowerOfTwo(n / k))                     return true;             }                // update k to next odd number             k = k + 2;         }            // If we reach here means         // there exists no value of K         // Such that k is odd number         // and n/k is a power of 2 greater than k         return false;     }        // Driver code     public static void Main()     {            // Get n         int n = 25;            // Check n for Proth Number         if (isProthNumber(n - 1))             Console.WriteLine("YES");         else             Console.WriteLine("NO");     } }

## PHP



Output:

YES

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