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Pythagorean Quadruple

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  • Difficulty Level : Hard
  • Last Updated : 22 Jun, 2022
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Given four points, check whether they form Pythagorean Quadruple. 
It is defined as a tuple of integers a, b, c, d such that a^2 + b^2 + c^2 = d^2   . They are basically the solutions of Diophantine Equations. In the geometric interpretation it represents a cuboid with integer side lengths |a|, |b|, |c| and whose space diagonal is |d| . 
 

The cuboids sides shown here are examples of pythagorean quadruples. 
It is primitive when their greatest common divisor is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. We can generate the set of primitive pythagorean quadruples for which a is odd can be generated by formula :
 

a = m2 + n2 – p2 – q2
b = 2(mq + np), 
c = 2(nq – mp), 
d = m2 + n2 + p2 + q2

where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q are odd. Thus, all primitive Pythagorean quadruples are characterized by Lebesgue’s identity.
 

(m2 + n2 + p2 + q2)2 = (2mq + 2nq)2 + 2(nq – mp)2 + (m2 + n2 – p2 – q2)m2 + n2 – p2 – q2

 

C++




// C++ code to detect Pythagorean Quadruples.
#include <bits/stdc++.h>
using namespace std;
 
// function for checking
bool pythagorean_quadruple(int a, int b, int c,
                                        int d)
{
    int sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
 
// Driver Code
int main()
{
    int a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        cout << "Yes" << endl;
    else
        cout << "No" << endl;
}

Java




// Java code to detect Pythagorean Quadruples.
import java.io.*;
import java.util.*;
 
class GFG {
 
// function for checking
static Boolean pythagorean_quadruple(int a, int b,
                                   int c, int d)
{
    int sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
 
// Driver function
    public static void main (String[] args) {
    int a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        System.out.println("Yes");
    else
        System.out.println("No" );
         
    }
}
// This code is contributed by Gitanjali.

Python3




# Python  code to detect
# Pythagorean Quadruples.
import math
 
# function for checking
def pythagorean_quadruple(a,b, c, d):
 
    sum = a * a + b * b + c * c;
    if (d * d == sum):
        return True
    else:
        return False
 
#driver code
a = 1
b = 2
c = 2
d = 3
if (pythagorean_quadruple(a, b, c, d)):
    print("Yes")
else:
     print("No" )
 
# This code is contributed
# by Gitanjali.

C#




// C# code to detect
// Pythagorean Quadruples.
using System;
 
class GFG {
 
    // function for checking
    static Boolean pythagorean_quadruple(int a,
                            int b, int c, int d)
    {
        int sum = a * a + b * b + c * c;
        if (d * d == sum)
            return true;
        else
            return false;
    }
     
    // Driver function
        public static void Main () {
             
        int a = 1, b = 2, c = 2, d = 3;
         
        if (pythagorean_quadruple(a, b, c, d))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No" );
             
    }
}
 
// This code is contributed by vt_M.

PHP




<?php
// php code to detect Pythagorean Quadruples.
 
// function for checking
function pythagorean_quadruple($a, $b, $c, $d)
{
    $sum = $a * $a + $b * $b + $c * $c;
     
    if ($d * $d == $sum)
        return true;
    else
        return false;
}
 
// Driver Code
    $a = 1; $b = 2; $c = 2; $d = 3;
     
    if (pythagorean_quadruple($a, $b, $c, $d))
        echo "Yes" ;
    else
        echo "No" ;
         
// This code is contributed by anuj_67.
?>

Javascript




<script>
 
// JavaScript program to detect Pythagorean Quadruples.
 
// function for checking
function pythagorean_quadruple(a, b,
                                   c, d)
{
    let sum = a * a + b * b + c * c;
    if (d * d == sum)
        return true;
    else
        return false;
}
       
 
// Driver code
         
        let a = 1, b = 2, c = 2, d = 3;
    if (pythagorean_quadruple(a, b, c, d))
        document.write("Yes");
    else
        document.write("No" );
 
</script>

Output: 

Yes

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

References 
Wiki 
mathworld
 


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