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# 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:

1. Iterate through values of k from 1 to n.
2. For each k, iterate through exponent from 0 to n.
3. Calculate prothNumber using the formula k * (1ULL << exponent) + 1.
4. If prothNumber is equal to the given number n, return true.
5. If prothNumber is greater than n, break the inner loop and continue with the next k.
6. If no Proth number is found, return false.

Note: This brute-force approach has high time complexity and is not efficient for large values of n.

## C++

 `#include ``#include ` `// Utility function to check if a number is a power of two``bool` `isPowerOfTwo(unsigned ``long` `long` `int` `n) {``    ``return` `n && !(n & (n - 1));``}` `// Function to check if a number is a Proth number using brute-force``bool` `isProthNumber(unsigned ``long` `long` `int` `n) {``    ``if` `(n < 3)``        ``return` `false``;` `    ``for` `(unsigned ``long` `long` `int` `k = 1; k <= n; ++k) {``        ``for` `(unsigned ``long` `long` `int` `exponent = 0; exponent <= n; ++exponent) {``            ``unsigned ``long` `long` `int` `prothNumber = k * (1ULL << exponent) + 1;``            ``if` `(prothNumber == n)``                ``return` `true``;``            ``if` `(prothNumber > n)``                ``break``;``        ``}``    ``}` `    ``return` `false``;``}``// Nikunj Sonigara``// Driver code``int` `main() {``    ``// Get n``    ``unsigned ``long` `long` `int` `n = 25;` `    ``// Check n for Proth Number``    ``if` `(isProthNumber(n - 1))``        ``std::cout << ``"YES"` `<< std::endl;``    ``else``        ``std::cout << ``"NO"` `<< std::endl;` `    ``return` `0;``}`

## Java

 `public` `class` `GFG {``    ` `    ``// Utility function to check if a number is a power of two``    ``public` `static` `boolean` `isPowerOfTwo(``long` `n) {``        ``return` `n != ``0` `&& (n & (n - ``1``)) == ``0``;``    ``}` `    ``// Function to check if a number is a Proth number using brute-force``    ``public` `static` `boolean` `isProthNumber(``long` `n) {``        ``if` `(n < ``3``)``            ``return` `false``;` `        ``for` `(``long` `k = ``1``; k <= n; ++k) {``            ``for` `(``long` `exponent = ``0``; exponent <= n; ++exponent) {``                ``long` `prothNumber = k * (1L << exponent) + ``1``;``                ``if` `(prothNumber == n)``                    ``return` `true``;``                ``if` `(prothNumber > n)``                    ``break``;``            ``}``        ``}` `        ``return` `false``;``    ``}``  ` `    ``// Nikunj Sonigara``    ``public` `static` `void` `main(String[] args) {``        ``// Get n``        ``long` `n = ``25``;` `        ``// Check n for Proth Number``        ``if` `(isProthNumber(n - ``1``))``            ``System.out.println(``"YES"``);``        ``else``            ``System.out.println(``"NO"``);``    ``}``}`

## Python3

 `# Utility function to check if a number is a power of two``def` `isPowerOfTwo(n):``    ``return` `n !``=` `0` `and` `(n & (n ``-` `1``)) ``=``=` `0` `# Function to check if a number is a Proth number using brute-force``def` `isProthNumber(n):``    ``if` `n < ``3``:``        ``return` `False` `    ``for` `k ``in` `range``(``1``, n``+``1``):``        ``for` `exponent ``in` `range``(``0``, n``+``1``):``            ``prothNumber ``=` `k ``*` `(``1` `<< exponent) ``+` `1``            ``if` `prothNumber ``=``=` `n:``                ``return` `True``            ``if` `prothNumber > n:``                ``break` `    ``return` `False` `# Nikunj Sonigara``# Driver code``n ``=` `25` `# Check n for Proth Number``if` `isProthNumber(n ``-` `1``):``    ``print``(``"YES"``)``else``:``    ``print``(``"NO"``)`

Output

`YES`

Time Complexity: O(N2)
Auxiliary Space: O(1)

Approach 2:

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 numbers 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 the 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 function 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

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## Javascript

 ``

Output

`YES`

Time Complexity: O(sqrt(n))
Auxiliary Space: O(1)

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