Pythagorean Triplet with given sum

The following code prints the pythagorean triplet if it is equal to the input, but the problem is that it takes a long time for large numbers like 90,000 to answer. What can I do to optimize the following code? 1 ≤ n ≤ 90 000

def pythagoreanTriplet(n):

    # Considering triplets in
    # sorted order. The value
    # of first element in sorted
    # triplet can be at-most n/3.
    for i in range(1, int(n / 3) + 1):

        # The value of second element
        # must be less than equal to n/2
        for j in range(i + 1,
                       int(n / 2) + 1):

            k = n - i - j
            if (i * i + j * j == k * k):
                print(i, ", ", j, ", ",
                      k, sep="")
                return

    print("Impossible")
# Driver Code
vorodi = int(input())
pythagoreanTriplet(vorodi)

Your source code does a brute force search for a solution so it's slow.

Faster Code

def solve_pythagorean_triplets(n):
  " Solves for triplets whose sum equals n "
  solutions = []
  for a in range(1, n):
    denom = 2*(n-a)
    num = 2*a**2 + n**2 - 2*n*a
    if denom > 0 and num % denom == 0:
      c = num // denom
      b = n - a - c
      if b > a:
        solutions.append((a, b, c))

  return solutions

OP code

Modified OP code so it returns all solutions rather than printing the first found to compare performance

def pythagoreanTriplet(n): 
  
    # Considering triplets in  
    # sorted order. The value  
    # of first element in sorted  
    # triplet can be at-most n/3. 
    results = []
    for i in range(1, int(n / 3) + 1):  
          
        # The value of second element  
        # must be less than equal to n/2 
        for j in range(i + 1,  
                       int(n / 2) + 1):  
  
            k = n - i - j 
            if (i * i + j * j == k * k):
                results.append((i, j, k))
      
    return results

Timing

 n     pythagoreanTriplet (OP Code)     solve_pythagorean_triplets (new)
  900   0.084 seconds                       0.039 seconds
  5000  3.130 seconds                       0.012 seconds
  90000 Timed out after several minutes     0.430 seconds

Explanation

Function solve_pythagorean_triplets is O(n) algorithm that works as follows.

1. Searching for:

   a^2 + b^2 = c^2 (triplet)
   a + b + c = n   (sum equals input)
   

2. Solve by searching over a (i.e. a fixed for an iteration). With a fixed, we have two equations and two unknowns (b, c):

   b + c = n - a
   c^2 - b^2 = a^2
   

3. Solution is:

   denom = 2*(n-a)
   num = 2*a**2 + n**2 - 2*n*a
   if denom > 0 and num % denom == 0:
       c = num // denom
       b = n - a - c
       if b > a:
           (a, b, c) # is a solution
   

4. Iterate a range(1, n) to get different solutions

Edit June 2022 by @AbhijitSarkar:

For those who like to see the missing steps:

c^2 - b^2 = a^2
b + c = n - a
=> b = n - a - c

c^2 - (n - a - c)^2 = a^2
=> c^2 - (n - a - c) * (n - a - c) = a^2
=> c^2 - n(n - a - c) + a(n - a - c) + c(n - a - c) = a^2
=> c^2 - n^2 + an + nc + an - a^2 - ac + cn - ac - c^2 = a^2
=> -n^2 + 2an + 2nc - a^2 - 2ac = a^2
=> -n^2 + 2an + 2nc - 2a^2 - 2ac = 0
=> 2c(n - a) = n^2 - 2an + 2a^2
=> c = (n^2 - 2an + 2a^2) / 2(n - a)