# 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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