Check if N can be represented as sum of squares of two consecutive integers

Given an integer N, the task is to check whether N can be represented as a sum of squares of two consecutive integers or not.

Examples:

Input: N = 5
Output: Yes
Explanation:
The integer 5 = 12 + 22 where 1 and 2 are consecutive numbers.

Input: 13
Output: Yes
Explanation:
13 = 22 + 32

Approach: This equation can be represented as:



=> N = K^{2} + (K - 1)^{2}

=> N = 2*K^{2} - 2*K + 1

=> K = \frac{2 + \sqrt{8*N - 4}}{2}

Finally, check the value of computed using this formula is an integer, which means that N can be represented as the sum of squares 2 consecutive integers

Below is the implementation of the above approach:

C++

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// C++ implementation to check that
// a number is sum of squares of 2
// consecutive numbers or not
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to check that the
// a number is sum of squares of 2
// consecutive numbers or not
bool isSumSquare(int N)
{
    float n
        = (2 + sqrt(8 * N - 4))
          / 2;
  
    // Condition to check if the
    // a number is sum of squares of 2
    // consecutive numbers or not
    return (n - (int)n) == 0;
}
  
// Driver Code
int main()
{
    int i = 13;
  
    // Function call
    if (isSumSquare(i)) {
        cout << "Yes";
    }
    else {
        cout << "No";
    }
    return 0;
}

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Java

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// Java implementation to check that 
// a number is sum of squares of 2 
// consecutive numbers or not 
import java.lang.Math;
  
class GFG{
      
// Function to check that the 
// a number is sum of squares of 2 
// consecutive numbers or not 
public static boolean isSumSquare(int N) 
    double n = (2 + Math.sqrt(8 * N - 4)) / 2
      
    // Condition to check if the 
    // a number is sum of squares of 2 
    // consecutive numbers or not 
    return(n - (int)n) == 0
  
// Driver code 
public static void main(String[] args)
{
    int i = 13
  
    // Function call 
    if (isSumSquare(i))
    
        System.out.println("Yes");
    
    else
    
        System.out.println("No");
    
}
}
  
// This code is contributed by divyeshrabadiya07

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Python3

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# Python3 implementation to check that 
# a number is sum of squares of 2 
# consecutive numbers or not 
import math
  
# Function to check that the a
# number is sum of squares of 2 
# consecutive numbers or not 
def isSumSquare(N):
  
    n = (2 + math.sqrt(8 * N - 4)) / 2
      
    # Condition to check if the a 
    # number is sum of squares of
    # 2 consecutive numbers or not
    return (n - int(n)) == 0
  
# Driver code 
if __name__=='__main__':
      
    i = 13
      
    # Function call
    if isSumSquare(i):
        print('Yes')
    else :
        print('No')
  
# This code is contributed by rutvik_56

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C#

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// C# implementation to check that 
// a number is sum of squares of 2 
// consecutive numbers or not 
using System;
class GFG{
      
// Function to check that the 
// a number is sum of squares of 2 
// consecutive numbers or not 
public static bool isSumSquare(int N) 
    double n = (2 + Math.Sqrt(8 * N - 4)) / 2; 
      
    // Condition to check if the 
    // a number is sum of squares of 2 
    // consecutive numbers or not 
    return(n - (int)n) == 0; 
  
// Driver code 
public static void Main(String[] args)
{
    int i = 13; 
  
    // Function call 
    if (isSumSquare(i))
    
        Console.WriteLine("Yes");
    
    else
    
        Console.WriteLine("No");
    
}
}
  
// This code is contributed by sapnasingh4991

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Output:

Yes

Note: In order to print the integers, we can easily solve the above equation to get the roots.

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