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

Deduct 1 from the number. This would give a number in the form k*2ⁿ, if the given number is a proth number.
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.
If found, print ‘YES’
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 <bits/stdc++.h> 
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

<?php 
// PHP program to check Proth number 
  
// Utility function to check  
// power of two 
function isPowerOfTwo($n) 
{ 
    return ($n && !($n & ($n - 1))); 
} 
  
// Function to check if the 
// Given number is Proth  
// number or not 
function 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 
$n = 25; 
  
// Check n for Proth Number 
if (isProthNumber($n - 1)) 
    echo "YES"; 
else
    echo "NO"; 
      
// This code is contributed  
// by inder_verma 
?> 

Output:

YES

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My Personal Notes

C++

Java

Python3

C#

PHP

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