# Pythagorean Triplet in an array

Given an array of integers, write a function that returns true if there is a triplet (a, b, c) that satisfies a2 + b2 = c2.

Example:

```Input: arr[] = {3, 1, 4, 6, 5}
Output: True
There is a Pythagorean triplet (3, 4, 5).

Input: arr[] = {10, 4, 6, 12, 5}
Output: False
There is no Pythagorean triplet.
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Method 1 (Naive)
A simple solution is to run three loops, three loops pick three array elements and check if current three elements form a Pythagorean Triplet.

Below is implementation of simple solution.

## C++

```// A C++ program that returns true if there is a Pythagorean
// Triplet in a given aray.
#include <iostream>
using namespace std;

// Returns true if there is Pythagorean triplet in ar[0..n-1]
bool isTriplet(int ar[], int n)
{
for (int i=0; i<n; i++)
{
for (int j=i+1; j<n; j++)
{
for (int k=j+1; k<n; k++)
{
// Calculate square of array elements
int x = ar[i]*ar[i], y = ar[j]*ar[j], z = ar[k]*ar[k];

if (x == y + z || y == x + z || z == x + y)
return true;
}
}
}

// If we reach here, no triplet found
return false;
}

/* Driver program to test above function */
int main()
{
int ar[] = {3, 1, 4, 6, 5};
int ar_size = sizeof(ar)/sizeof(ar[0]);
isTriplet(ar, ar_size)? cout << "Yes": cout << "No";
return 0;
}```

## Java

```// A Java program that returns true if there is a Pythagorean
// Triplet in a given aray.
import java.io.*;

class PythagoreanTriplet {

// Returns true if there is Pythagorean triplet in ar[0..n-1]
static boolean isTriplet(int ar[], int n)
{
for (int i=0; i<n; i++)
{
for (int j=i+1; j<n; j++)
{
for (int k=j+1; k<n; k++)
{
// Calculate square of array elements
int x = ar[i]*ar[i], y = ar[j]*ar[j], z = ar[k]*ar[k];

if (x == y + z || y == x + z || z == x + y)
return true;
}
}
}

// If we reach here, no triplet found
return false;
}

// Driver program to test above function
public static void main(String[] args)
{
int ar[] = {3, 1, 4, 6, 5};
int ar_size = ar.length;
if(isTriplet(ar,ar_size)==true)
System.out.println("Yes");
else
System.out.println("No");
}
}
/* This code is contributed by Devesh Agrawal */
```

## Python

```# Python program to check if there is Pythagorean
# triplet in given array

# Returns true if there is Pythagorean
# triplet in ar[0..n-1]
def isTriplet(ar, n):
for i in range(n - 2):
for k in range(j + 1, n):
for j in range(i + 1, n - 1):
# Calculate square of array elements
x = ar[i]*ar[i]
y = ar[j]*ar[j]
z = ar[k]*ar[k]
if (x == y + z or y == x + z or z == x + y):
return 1

# If we reach here, no triplet found
return 0

# Driver program to test above function
ar = [3, 1, 4, 6, 5]
ar_size = len(ar)

if(isTriplet(ar, ar_size)):
print("Yes")
else:
print("No")

# This code is contributed by Aditi Sharma
```

Output:
`Yes`

Time Complexity of the above solution is O(n3).

Method 2 (Use Sorting)
We can solve this in O(n2) time by sorting the array first.

1) Do square of every element in input array. This step takes O(n) time.

2) Sort the squared array in increasing order. This step takes O(nLogn) time.

3) To find a triplet (a, b, c) such that a = b + c, do following.

1. Fix ‘a’ as last element of sorted array.
2. Now search for pair (b, c) in subarray between first element and ‘a’. A pair (b, c) with given sum can be found in O(n) time using meet in middle algorithm discussed in method 1 of this post.
3. If no pair found for current ‘a’, then move ‘a’ one position back and repeat step 3.2.

Below is C++ implementation of above algorithm.

## C++

```// A C++ program that returns true if there is a Pythagorean
// Triplet in a given array.
#include <iostream>
#include <algorithm>
using namespace std;

// Returns true if there is a triplet with following property
// A[i]*A[i] = A[j]*A[j] + A[k]*[k]
// Note that this function modifies given array
bool isTriplet(int arr[], int n)
{
// Square array elements
for (int i=0; i<n; i++)
arr[i] = arr[i]*arr[i];

// Sort array elements
sort(arr, arr + n);

// Now fix one element one by one and find the other two
// elements
for (int i = n-1; i >= 2; i--)
{
// To find the other two elements, start two index
// variables from two corners of the array and move
// them toward each other
int l = 0; // index of the first element in arr[0..i-1]
int r = i-1; // index of the last element in arr[0..i-1]
while (l < r)
{
// A triplet found
if (arr[l] + arr[r] == arr[i])
return true;

// Else either move 'l' or 'r'
(arr[l] + arr[r] < arr[i])?  l++: r--;
}
}

// If we reach here, then no triplet found
return false;
}

/* Driver program to test above function */
int main()
{
int arr[] = {3, 1, 4, 6, 5};
int arr_size = sizeof(arr)/sizeof(arr[0]);
isTriplet(arr, arr_size)? cout << "Yes": cout << "No";
return 0;
}
```

## Java

```// A Java program that returns true if there is a Pythagorean
// Triplet in a given aray.
import java.io.*;
import java.util.*;

class PythagoreanTriplet
{
// Returns true if there is a triplet with following property
// A[i]*A[i] = A[j]*A[j] + A[k]*[k]
// Note that this function modifies given array
static boolean isTriplet(int arr[], int n)
{
// Square array elements
for (int i=0; i<n; i++)
arr[i] = arr[i]*arr[i];

// Sort array elements
Arrays.sort(arr);

// Now fix one element one by one and find the other two
// elements
for (int i = n-1; i >= 2; i--)
{
// To find the other two elements, start two index
// variables from two corners of the array and move
// them toward each other
int l = 0; // index of the first element in arr[0..i-1]
int r = i-1; // index of the last element in arr[0..i-1]
while (l < r)
{
// A triplet found
if (arr[l] + arr[r] == arr[i])
return true;

// Else either move 'l' or 'r'
if (arr[l] + arr[r] < arr[i])
l++;
else
r--;
}
}

// If we reach here, then no triplet found
return false;
}

// Driver program to test above function
public static void main(String[] args)
{
int arr[] = {3, 1, 4, 6, 5};
int arr_size = arr.length;
if (isTriplet(arr,arr_size)==true)
System.out.println("Yes");
else
System.out.println("No");
}
}
/*This code is contributed by Devesh Agrawal*/
```

## Python

```# Python program that returns true if there is
# a Pythagorean Triplet in a given array.

# Returns true if there is Pythagorean
# triplet in ar[0..n-1]
def isTriplet(ar, n):
# Square all the elemennts
for i in range(n):
ar[i] = ar[i] * ar[i]

# sort array elements
ar.sort()

# fix one element
# and find other two
# i goes from n - 1 to 2
for i in range(n-1, 1, -1):
# start two index variables from
# two corners of the array and
# move them toward each other
j = 0
k = i - 1
while (j < k):
# A triplet found
if (ar[j] + ar[k] == ar[i]):
return True
else:
if (ar[j] + ar[k] < ar[i]):
j = j + 1
else:
k = k - 1
# If we reach here, then no triplet found
return False

# Driver program to test above function */
ar = [3, 1, 4, 6, 5]
ar_size = len(ar)
if(isTriplet(ar, ar_size)):
print("Yes")
else:
print("No")

# This code is contributed by Aditi Sharma
```

Output:

` Yes `

Time complexity of this method is O(n2).

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