Pythagorean Quadruple

A group of four positive integers (a, b, c, and d) that satisfy the Pythagorean equation are called Pythagorean quadruples.

The equation can be written as: a² + b² + c² = d² , with ‘d’ being the largest value out of the given numbers. In other words, the square of the fourth integer should be equal to the sum obtained by adding the squares of the previous three numbers.

(1, 2, 2, 3) is a Pythagorean quadruplet as (1² + 2² + 2²) = (1 + 4 + 4) = (9) = (3²). Due to the requirement that they satisfy the Pythagorean equation, not all sets of four positive integers can be Pythagorean quadruples. Pythagorean quadruples can be used to create intriguing mathematical patterns and have applications in number theory as well.

Problem Statement

The aim of this problem is to check whether the 4 given input integers form a Pythagorean Quadruple.

Examples

Input: {2, 3, 6, 7}
Output: True

Explanation −

The sum of the squares of the first three numbers can be written as,

(2² + 3² + 6²) = (4 + 9 + 36) = 49

The square of the fourth number is 72 = 49, which is equal to the previous result. Hence the numbers, {2, 3, 6, 7}, form a Pythagorean Quadruplet so we return True.

Input: {1, 4, 8, 9}
Output: True

Explanation −

The sum of the squares of the first three numbers can be written as,

(1² + 4² + 8²) = (1 + 16 + 64) = 81

The square of the fourth number is 92 = 81, which is equal to the previous result. Hence the numbers, {1, 4, 8, 9}, form a Pythagorean Quadruplet so we return True.

Input: {1, 2, 3, 4}
Output: False

Explanation −

The sum of the squares of the first three numbers can be written as,

(1² + 2² + 3²) = (1 + 4 + 9) = 14

The square of the fourth number is 42 = 16, which is not equal to the previous result. Hence the numbers, {1, 4, 8, 9}, do not form a Pythagorean Quadruplet so we return False.

Solution Approach

In order to determine whether the 4 given numbers form a Pythagorean Quadruplet, we must figure out if the summation of the squares of the first 3 numbers is equal to the square of the fourth number, where the fourth number has the largest magnitude.

Algorithm

The approach consists of the following steps −

• Sort the vector of numbers in non-decreasing order.

• Compute the sum of squares of the first three numbers.

• Compute the square of the fourth number.

• Check if the results obtained from step 2 and step 3 are equal.

• Display the answer.

Pseudo Code

Function is_pythagorean_quadruplet()

• Initialise squareSum.

• squareSum = a² + b² + c²

• Return (squareSum == d²)

Function main()

• Initialise vector<int>nums = {a, b, c, d}

• sort (nums.begin(), nums.end())

• if (is_pythagorean_quadruplet())

• return true

• else

• return false

• Print output

Example: C++ Program

This program code checks if the 4 given numbers, stored in a vector, form a pythagorean quadruplet using the is_pythagorean_quadruplet() boolean function. It initially sorts the vector using the C++ standard template library sort() function. It then computes the sum of squares of the first three numbers and stores them in the squareSum variable. The function is_pythagorean_quadruplet() returns true if the value of squareSum is equal to the square of the fourth number else it returns false.

// C++ code for to check if the 4 given numbers form a Pythagorean Quadruple
#include <iostream>
#include <cmath>
#include <vector>
#include <algorithm>
using namespace std;

// this function is used to check out whether the 4 given numbers form a Pythagorean Quadruplet or not
bool is_pythagorean_quadruplet(vector<int> nums){
   int squareSum;
   squareSum = pow(nums[0], 2) + pow(nums[1], 2) + pow(nums[2], 2); // 9
   return (squareSum == pow(nums[3], 2)); // compares summation of square   of  first three numbers with square of fourth number (9 == 9)
}

int main(){
   vector<int> nums = {1, 2, 2, 3};
   sort(nums.begin(), nums.end()); // sorts the numbers in non-decreasing order
   if (is_pythagorean_quadruplet(nums)) {
      cout << "True" << endl; 
   } else {
      cout << "False" << endl;
   }
   return 0;
}

Output

True

Time and Space Complexity Analysis

Time Complexity: O(1)

O(1), since it does a fixed amount of operations. The squareSum is calculated using three constant-time operations by the function is_pythagorean_quadruplet(), and its equality with the fourth element raised to the power of two is tested.

In the worst scenario, the sort function's time complexity is O(nlog(n)). As a result, the overall time complexity of the code should be O(nlog(n)), where n is the number of elements in the vector. Yet, the time complexity of the sort function remains constant because the size of the vector is set at 4 in each case

Space Complexity: O(1)

This is because the space required to store the input vector is always 4. No auxiliary space is needed in the implementation of the code; therefore the space complexity can be considered constant.

Conclusion

In conclusion, a Pythagorean Quadruple is a tuple of integers a, b, c, and d such that

a² + b² + c² = d². The algorithm presented in this article provides a straightforward and efficient way to determine if a tuple of four integers forms a Pythagorean quadruplet in constant time without the use of auxiliary space. The article discusses the basic concept of Pythagorean Quadruples along with suitable examples. It also provides the solution approach, the algorithm used and the C++ program solution as well, along with an in depth analysis of its time complexity and space complexity.