Generate Pythagorean Triplets

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++

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// 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_limitit
    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;
}

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Java

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// 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_limitit
        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.

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Python3

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# 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.

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C#

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// 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_limitit
        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.

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PHP

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<?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_limitit
    $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.
?>

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Output :

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)

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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