Generate Pythagorean Triplets
Last Updated :
23 Jul, 2025
Given a positive integer limit, your task is to find all possible Pythagorean Triplet (a, b, c), such that a <= b <= c <= limit.
Note: A Pythagorean triplet is a set of three positive integers a, b, and c such that a2 + b2 = c2.
Input: limit = 20
Output: 3 4 5
5 12 13
6 8 10
8 15 17
9 12 15
12 16 20
Explanation: All the triplets are arranged in the format (a, b, c), where a2 + b2 = c2.
[Naive Approach] - Using Nested Loops - O(n ^ 3) Time and O(1) Space
The idea is to use nested loops to generate all possible combinations of a, b, and c. For each combination check if a2 + b2 = c2, if the condition satisfies store the triplet, else move to next combination.
C++
// C++ program to find all Pythagorean
// Triplets smaller than a given limit
#include<bits/stdc++.h>
using namespace std;
// Function to generate all Pythagorean
// Triplets smaller than limit
vector<vector<int>> pythagoreanTriplets(int limit) {
// to hold the triplets
vector<vector<int>> ans;
for(int a = 1; a<=limit; a++) {
for(int b = a; b<=limit; b++) {
for(int c = b; c<=limit; c++) {
if(a * a + b * b == c * c) {
ans.push_back({a,b,c});
}
}
}
}
return ans;
}
int main() {
int limit = 20;
vector<vector<int>> ans = pythagoreanTriplets(limit);
for (int i = 0; i < ans.size(); i++) {
for (int j = 0; j < 3; j++) {
cout << ans[i][j] << " ";
}
cout << endl;
}
return 0;
}
// C program to find all Pythagorean
// Triplets smaller than a given limit
#include <stdio.h>
void pythagoreanTriplets(int limit) {
// to hold the triplets
int a, b, c;
for(a = 1; a <= limit; a++) {
for(b = a; b <= limit; b++) {
for(c = b; c <= limit; c++) {
if(a * a + b * b == c * c) {
printf("%d %d %d\n", a, b, c);
}
}
}
}
}
int main() {
int limit = 20;
pythagoreanTriplets(limit);
return 0;
}
// Java program to find all Pythagorean
// Triplets smaller than a given limit
import java.util.*;
class GFG {
// Function to generate all Pythagorean
// Triplets smaller than limit
static List<List<Integer>> pythagoreanTriplets(int limit) {
// to hold the triplets
List<List<Integer>> ans = new ArrayList<>();
for(int a = 1; a <= limit; a++) {
for(int b = a; b <= limit; b++) {
for(int c = b; c <= limit; c++) {
if(a * a + b * b == c * c) {
ans.add(Arrays.asList(a, b, c));
}
}
}
}
return ans;
}
public static void main(String[] args) {
int limit = 20;
List<List<Integer>> ans = pythagoreanTriplets(limit);
for (List<Integer> triplet : ans) {
for (int num : triplet) {
System.out.print(num + " ");
}
System.out.println();
}
}
}
# Python program to find all Pythagorean
# Triplets smaller than a given limit
def pythagoreanTriplets(limit):
# to hold the triplets
ans = []
for a in range(1, limit + 1):
for b in range(a, limit + 1):
for c in range(b, limit + 1):
if a * a + b * b == c * c:
ans.append([a, b, c])
return ans
# Driver code
limit = 20
ans = pythagoreanTriplets(limit)
for triplet in ans:
for num in triplet:
print(num, end=" ")
print()
// C# program to find all Pythagorean
// Triplets smaller than a given limit
using System;
using System.Collections.Generic;
class GFG {
// Function to generate all Pythagorean
// Triplets smaller than limit
static List<List<int>> pythagoreanTriplets(int limit) {
// to hold the triplets
List<List<int>> ans = new List<List<int>>();
for(int a = 1; a <= limit; a++) {
for(int b = a; b <= limit; b++) {
for(int c = b; c <= limit; c++) {
if(a * a + b * b == c * c) {
ans.Add(new List<int> { a, b, c });
}
}
}
}
return ans;
}
static void Main() {
int limit = 20;
List<List<int>> ans = pythagoreanTriplets(limit);
foreach (List<int> triplet in ans) {
foreach (int num in triplet) {
Console.Write(num + " ");
}
Console.WriteLine();
}
}
}
// JavaScript program to find all Pythagorean
// Triplets smaller than a given limit
function pythagoreanTriplets(limit) {
// to hold the triplets
let ans = [];
for(let a = 1; a <= limit; a++) {
for(let b = a; b <= limit; b++) {
for(let c = b; c <= limit; c++) {
if(a * a + b * b === c * c) {
ans.push([a, b, c]);
}
}
}
}
return ans;
}
// Driver code
let limit = 20;
let ans = pythagoreanTriplets(limit);
for (let triplet of ans) {
console.log(triplet.join(" "));
}
Output3 4 5
5 12 13
6 8 10
8 15 17
9 12 15
12 16 20
Time Complexity: O(n3), where n is representing the limit.
Space Complexity: O(1)
[Expected Approach] - Using Two Pointers - O(n ^ 2) Time and O(1) Space
The idea is to use Two Pointers to generate combinations of a & b in linear time, and check if a2 + b2 = c2. To do so, iterate through all values of c from 1 to limit, and set a = 1, and b = c - 1. At each iteration there are three possibilities:
- if a2 + b2 > c2: Decrease the value of b by 1.
- if a2 + b2 < c2: Increase the value of a by 1.
- if a2 + b2 = c2: Store the triplet
C++
// C++ program to find all Pythagorean
// Triplets smaller than a given limit
#include<bits/stdc++.h>
using namespace std;
// Function to generate all Pythagorean
// Triplets smaller than limit
vector<vector<int>> pythagoreanTriplets(int limit) {
// to hold the triplets
vector<vector<int>> ans;
for(int c = 1; c<=limit; c++) {
int a = 1, b = c - 1;
while (a <= b) {
if (a * a + b * b == c * c) {
ans.push_back({a,b,c});
break;
}
else if (a * a + b * b < c * c) {
a++;
}
else {
b--;
}
}
}
return ans;
}
int main() {
int limit = 20;
vector<vector<int>> ans = pythagoreanTriplets(limit);
for (int i = 0; i < ans.size(); i++) {
for (int j = 0; j < 3; j++) {
cout << ans[i][j] << " ";
}
cout << endl;
}
return 0;
}
// Java program to find all Pythagorean
// Triplets smaller than a given limit
import java.util.*;
class GFG {
// Function to generate all Pythagorean
// Triplets smaller than limit
static List<List<Integer>> pythagoreanTriplets(int limit) {
// to hold the triplets
List<List<Integer>> ans = new ArrayList<>();
for(int c = 1; c <= limit; c++) {
int a = 1, b = c - 1;
while (a <= b) {
if (a * a + b * b == c * c) {
ans.add(Arrays.asList(a, b, c));
break;
}
else if (a * a + b * b < c * c) {
a++;
}
else {
b--;
}
}
}
return ans;
}
public static void main(String[] args) {
int limit = 20;
List<List<Integer>> ans = pythagoreanTriplets(limit);
for (int i = 0; i < ans.size(); i++) {
for (int j = 0; j < 3; j++) {
System.out.print(ans.get(i).get(j) + " ");
}
System.out.println();
}
}
}
# Python program to find all Pythagorean
# Triplets smaller than a given limit
def pythagoreanTriplets(limit):
# to hold the triplets
ans = []
for c in range(1, limit + 1):
a, b = 1, c - 1
while a <= b:
if a * a + b * b == c * c:
ans.append([a, b, c])
break
elif a * a + b * b < c * c:
a += 1
else:
b -= 1
return ans
# Driver code
limit = 20
ans = pythagoreanTriplets(limit)
for i in range(len(ans)):
for j in range(3):
print(ans[i][j], end=" ")
print()
// C# program to find all Pythagorean
// Triplets smaller than a given limit
using System;
using System.Collections.Generic;
class GFG {
// Function to generate all Pythagorean
// Triplets smaller than limit
static List<List<int>> pythagoreanTriplets(int limit) {
// to hold the triplets
List<List<int>> ans = new List<List<int>>();
for(int c = 1; c <= limit; c++) {
int a = 1, b = c - 1;
while (a <= b) {
if (a * a + b * b == c * c) {
ans.Add(new List<int> { a, b, c });
break;
}
else if (a * a + b * b < c * c) {
a++;
}
else {
b--;
}
}
}
return ans;
}
public static void Main() {
int limit = 20;
List<List<int>> ans = pythagoreanTriplets(limit);
for (int i = 0; i < ans.Count; i++) {
for (int j = 0; j < 3; j++) {
Console.Write(ans[i][j] + " ");
}
Console.WriteLine();
}
}
}
// JavaScript program to find all Pythagorean
// Triplets smaller than a given limit
function pythagoreanTriplets(limit) {
// to hold the triplets
let ans = [];
for(let c = 1; c <= limit; c++) {
let a = 1, b = c - 1;
while (a <= b) {
if (a * a + b * b === c * c) {
ans.push([a, b, c]);
break;
}
else if (a * a + b * b < c * c) {
a++;
}
else {
b--;
}
}
}
return ans;
}
// Driver code
let limit = 20;
let ans = pythagoreanTriplets(limit);
for (let i = 0; i < ans.length; i++) {
for (let j = 0; j < 3; j++) {
process.stdout.write(ans[i][j] + " ");
}
console.log();
}
Output3 4 5
6 8 10
5 12 13
9 12 15
8 15 17
12 16 20
Time Complexity: O(n2), where n is representing the triplets.
Space Complexity: O(1)
[Alternate Approach] - Using Mathematics
Note: The below given 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.
The idea is to use square sum relation of Pythagorean Triplet that states that addition of squares of a and b is equal to square of c. We can write these numbers 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.
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 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 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# 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.
<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>
<?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.
?>
Output3 4 5
8 6 10
5 12 13
15 8 17
12 16 20
Time complexity of this approach is O(√n) where n is the given limit. We are iterating until c <= limit ,where c = m^2 + n^2. Thus the iteration will take place approximately sqrt(limit) times.
Auxiliary space: O(1) as it is using constant space for variables
References:
https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples
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