A Pythagorean triplet is a set of three positive integers a, b and c such that a2 + b2 = c2. Given a limit, generate all Pythagorean Triples with values smaller than given limit.
Input : limit = 20
Output : 3 4 5
8 6 10
5 12 13
15 8 17
12 16 20
A Simple Solution is to generate these triplets smaller than given limit using three nested loop. For every triplet, check if Pythagorean condition is true, if true, then print the triplet. Time complexity of this solution is O(limit3) where ‘limit’ is given limit.
An Efficient Solution can print all triplets in O(k) time where k is number of triplets printed. The idea is to use square sum relation of Pythagorean triplet, i.e., addition of squares of a and b is equal to square of c, we can write these number in terms of m and n such that,
a = m2 - n2
b = 2 * m * n
c = m2 + n2
because,
a2 = m4 + n4 – 2 * m2 * n2
b2 = 4 * m2 * n2
c2 = m4 + n4 + 2* m2 * n2
We can see that a2 + b2 = c2, so instead of iterating for a, b and c we can iterate for m and n and can generate these triplets.
Below is the implementation of above idea :
C++
// C++ program to generate pythagorean // triplets smaller than a given limit #include <bits/stdc++.h> // Function to generate pythagorean // triplets smaller than limit void pythagoreanTriplets(int limit) { // triplet: a^2 + b^2 = c^2 int a, b, c = 0; // loop from 2 to max_limit int m = 2; // Limiting c would limit // all a, b and c while (c < limit) { // now loop on j from 1 to i-1 for (int n = 1; n < m; ++n) { // Evaluate and print triplets using // the relation between a, b and c a = m * m - n * n; b = 2 * m * n; c = m * m + n * n; if (c > limit) break; printf("%d %d %d\n", a, b, c); } m++; } } // Driver Code int main() { int limit = 20; pythagoreanTriplets(limit); return 0; } |
Java
// Java program to generate pythagorean // triplets smaller than a given limit import java.io.*; import java.util.*; class GFG { // Function to generate pythagorean // triplets smaller than limit static void pythagoreanTriplets(int limit) { // triplet: a^2 + b^2 = c^2 int a, b, c = 0; // loop from 2 to max_limit int m = 2; // Limiting c would limit // all a, b and c while (c < limit) { // now loop on j from 1 to i-1 for (int n = 1; n < m; ++n) { // Evaluate and print // triplets using // the relation between // a, b and c a = m * m - n * n; b = 2 * m * n; c = m * m + n * n; if (c > limit) break; System.out.println(a + " " + b + " " + c); } m++; } } // Driver Code public static void main(String args[]) { int limit = 20; pythagoreanTriplets(limit); } } // This code is contributed by Manish. |
Python3
# Python3 program to generate pythagorean # triplets smaller than a given limit # Function to generate pythagorean # triplets smaller than limit def pythagoreanTriplets(limits) : c, m = 0, 2 # Limiting c would limit # all a, b and c while c < limits : # Now loop on n from 1 to m-1 for n in range(1, m) : a = m * m - n * n b = 2 * m * n c = m * m + n * n # if c is greater than # limit then break it if c > limits : break print(a, b, c) m = m + 1 # Driver Code if __name__ == '__main__' : limit = 20 pythagoreanTriplets(limit) # This code is contributed by Shrikant13. |
C#
// C# program to generate pythagorean // triplets smaller than a given limit using System; class GFG { // Function to generate pythagorean // triplets smaller than limit static void pythagoreanTriplets(int limit) { // triplet: a^2 + b^2 = c^2 int a, b, c = 0; // loop from 2 to max_limit int m = 2; // Limiting c would limit // all a, b and c while (c < limit) { // now loop on j from 1 to i-1 for (int n = 1; n < m; ++n) { // Evaluate and print // triplets using // the relation between // a, b and c a = m * m - n * n; b = 2 * m * n; c = m * m + n * n; if (c > limit) break; Console.WriteLine(a + " " + b + " " + c); } m++; } } // Driver Code public static void Main() { int limit = 20; pythagoreanTriplets(limit); } } // This code is contributed by anuj_67. |
PHP
<?php // PHP program to generate pythagorean // triplets smaller than a given limit // Function to generate pythagorean // triplets smaller than limit function pythagoreanTriplets($limit) { // triplet: a^2 + b^2 = c^2 $a; $b; $c=0; // loop from 2 to max_limit $m = 2; // Limiting c would limit // all a, b and c while ($c < $limit) { // now loop on j from 1 to i-1 for ($n = 1; $n < $m; ++$n) { // Evaluate and print // triplets using the // relation between a, // b and c $a = $m *$m - $n * $n; $b = 2 * $m * $n; $c = $m * $m + $n * $n; if ($c > $limit) break; echo $a, " ", $b, " ", $c, "\n"; } $m++; } } // Driver Code $limit = 20; pythagoreanTriplets($limit); // This code is contributed by ajit. ?> |
Javascript
<script> // Javascript program to generate pythagorean // triplets smaller than a given limit // Function to generate pythagorean // triplets smaller than limit function pythagoreanTriplets(limit) { // Triplet: a^2 + b^2 = c^2 let a, b, c = 0; // Loop from 2 to max_limit let m = 2; // Limiting c would limit // all a, b and c while (c < limit) { // Now loop on j from 1 to i-1 for(let n = 1; n < m; ++n) { // Evaluate and print // triplets using // the relation between // a, b and c a = m * m - n * n; b = 2 * m * n; c = m * m + n * n; if (c > limit) break; document.write(a + " " + b + " " + c + "</br>"); } m++; } } // Driver code let limit = 20; pythagoreanTriplets(limit); // This code is contributed by divyesh072019 </script> |
3 4 5 8 6 10 5 12 13 15 8 17 12 16 20
Time complexity of this approach is O(k) where k is number of triplets printed for a given limit (We iterate for m and n only and every iteration prints a triplet)
Auxiliary space: O(1) as it is using constant space for variables
Note: The above method doesn’t generate all triplets smaller than a given limit. For example “9 12 15” which is a valid triplet is not printed by above method. Thanks to Sid Agrawal for pointing this out.
References:
https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples
This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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