Check if a given number is Pronic | Efficient Approach

A pronic number is such a number which can be represented as a product of two consecutive positive integers. By multiplying these two consecutive positive integers, there can be formed a rectangle which is represented by the product or pronic number. So it is also known as Rectangular Number.

The first few Pronic numbers are:
0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 . . . . . .

Pronic number is a number which is the product of two consecutive integers, that is, a number n is a product of x and (x+1). The task is to check if a given number is pronic or not.

Mathematical Representation:

If x is a pronic number, then x=n(n+1) ∀ n∈N0
Where, N0={0, 1, 2, 3, 4, ....}, (A set of Naturral Numbers)

Examples:

Input : 56
Output : YES
Explanation: 56 = 7 * 8 i.e 56 is a product 
of two consecutive integers 7 and 8.

Input : 65
Output : NO
Explanation: 65 cannot be represented as a
product of any two consecutive integers.

We had previously discussed an approach to check if a number is pronic or not in this article using a loop. The time Complexity of the previous algorithm is comparatively very high and in terms of Big-O asymptotic notation, it is O(√n).
In this article, we are going to explain an efficient approach with time complexity of O(log(log n). The idea is to observe that if a number can be expressed as the product of two consecutive integers then the two integers will be close to the square of root of that number. A more proper observation will lead to the fact that a number N can be represented as product of two consecutive integers only if the product of floor(sqrt(N)) and floor(sqrt(N))+1 is equal to N.

Below is the step by step algorithm of above approach:

Step 1: Evaluate the square root value of the given number.
Step 2: Calculate the floor value of that square root.
Step 3: Calculate the product of value calculated in step-2
	and its next consecutive number.
Step 4: Check the product value in step-3 with the given number.
	Step 4.1: If the condition satisfies,
		  then the number is a pronic number.
	Step 4.2: Otherwise the number is not a pronic number.

Below is the implementation of above algorithm:

C

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// C/C++ program to check if a number is pronic or not
  
#include<bits/stdc++.h>
using namespace std;
  
// function to check Pronic Number
bool pronic_check(int n)
{
    int x = (int)(sqrt(n));
  
    // Checking Pronic Number by 
    // multiplying consecutive numbers
    if (x*(x+1)==n) 
        return true;
    else
        return false;
}
  
// Driver Code
int main(void)
{
    int n = 56;    
    pronic_check(n) == true? cout << "YES"
                             cout << "NO";
      
    return 0;
}

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Java

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// Java program to check if a number is pronic or not
  
import java.io.*;
import java.util.*;
import java.math.*;
  
class GFG 
{
  
    // Function to check Pronic Number
    static boolean pronic_check(int n) 
    {
        int x = (int)(Math.sqrt(n));
      
        // Checking Pronic Number by 
        // multiplying consecutive numbers
        if (x * (x + 1) == n)
            return true;
        else
            return false;
    }
      
    // Driver Code
    public static void main(String[] args) 
    {
        int n = 56;        
        if (pronic_check(n)==true)
            System.out.println("YES");
        else
            System.out.println("NO");
    }
}

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Python3

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# Python program to check if a number is pronic or not
  
import math
  
# function to check Pronic Number
def pronic_check(n) :
    x = (int)(math.sqrt(n))
  
    # Checking Pronic Number by multiplying 
    # consecutive numbers
    if (x*(x + 1)== n):
        return True
    else:
        return False
  
# Driver Code
n = 56
  
if (pronic_check(n)==True):
    print("YES")
else:
    print("NO")

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C#

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// C# program to check if a number is 
// pronic or not
using System;
  
class GFG 
{
  
    // Function to check Pronic Number
    static bool pronic_check(int n) 
    {
        int x = (int)(Math.Sqrt(n));
      
        // Checking Pronic Number by 
        // multiplying consecutive numbers
        if (x * (x + 1) == n)
            return true;
        else
            return false;
    }
      
    // Driver Code
    public static void Main() 
    {
        int n = 56; 
          
        if (pronic_check(n)==true)
            Console.Write("YES");
        else
            Console.Write("NO");
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP program to check if a 
// number is pronic or not
  
// function to check Pronic Number
function pronic_check($n)
{
    $x = floor(sqrt($n));
  
    // Checking Pronic Number by 
    // multiplying consecutive numbers
    if ($x * ($x + 1) == $n
        return true;
    else
        return false;
}
  
    // Driver Code
    $n = 56; 
    if (pronic_check($n) == true)
        echo "YES" ;
    else
        echo "NO"
          
// This code is contributed by Sam007
?>

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Output:

YES


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Improved By : Sam007



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