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Python Program to Find Pythagorean Triplets

Intermediate · Numbers

A Pythagorean triplet is a set of three whole numbers a, b and c where a^2 + b^2 = c^2, such as 3, 4 and 5 (9 + 16 = 25). They are the sides of right-angled triangles, and this program finds every triplet whose sides are no bigger than a limit.

Program

pythagorean_triplets.py
import math
limit = 30 # or: limit = int(input("Largest side to check: "))
count = 0
print(f"Pythagorean triplets with sides up to {limit}:")
for a in range(1, limit + 1):
for b in range(a, limit + 1): # b starts at a, so (4, 3, 5) is not repeated
c_squared = a * a + b * b
c = math.isqrt(c_squared)
if c <= limit and c * c == c_squared:
print(f"{a}^2 + {b}^2 = {c}^2 ({a * a} + {b * b} = {c * c})")
count += 1
print("Triplets found:", count)
Output
Pythagorean triplets with sides up to 30:
3^2 + 4^2 = 5^2   (9 + 16 = 25)
5^2 + 12^2 = 13^2   (25 + 144 = 169)
6^2 + 8^2 = 10^2   (36 + 64 = 100)
7^2 + 24^2 = 25^2   (49 + 576 = 625)
8^2 + 15^2 = 17^2   (64 + 225 = 289)
9^2 + 12^2 = 15^2   (81 + 144 = 225)
10^2 + 24^2 = 26^2   (100 + 576 = 676)
12^2 + 16^2 = 20^2   (144 + 256 = 400)
15^2 + 20^2 = 25^2   (225 + 400 = 625)
18^2 + 24^2 = 30^2   (324 + 576 = 900)
20^2 + 21^2 = 29^2   (400 + 441 = 841)
Triplets found: 11

How it works

  1. The two loops try every pair of sides a and b, with b never smaller than a so each triplet appears only once.
  2. For each pair, a * a + b * b is the value that c * c would need to be.
  3. math.isqrt() gives the whole-number square root, and squaring it again shows whether the value really is a perfect square.
  4. If it is, and c is within the limit, the three numbers form a triplet and are printed with the working.

Other ways to write it

Euclid's formula

Euclid's formula
import math
print("Primitive triplets from Euclid's formula:")
for m in range(2, 6):
for n in range(1, m):
# m and n must share no factor, and one must be even
if math.gcd(m, n) == 1 and (m - n) % 2 == 1:
a = m * m - n * n
b = 2 * m * n
c = m * m + n * n
print(f"m={m}, n={n}: ({a}, {b}, {c})")
Output
Primitive triplets from Euclid's formula:
m=2, n=1: (3, 4, 5)
m=3, n=2: (5, 12, 13)
m=4, n=1: (15, 8, 17)
m=4, n=3: (7, 24, 25)
m=5, n=2: (21, 20, 29)
m=5, n=4: (9, 40, 41)

Euclid's formula builds triplets directly instead of searching. These are the primitive ones, with no common factor; multiplying one by 2, 3 and so on gives the rest, like 6, 8, 10.

Checking three given numbers

Checking three given numbers
def is_triplet(x, y, z):
a, b, c = sorted([x, y, z]) # the largest number must be c
return a * a + b * b == c * c
for sides in [(3, 4, 5), (13, 5, 12), (2, 3, 4)]:
print(sides, "is a triplet" if is_triplet(*sides) else "is not a triplet")
Output
(3, 4, 5) is a triplet
(13, 5, 12) is a triplet
(2, 3, 4) is not a triplet

Sorting first means the numbers can be given in any order, and *sides passes the three values in a tuple as three separate arguments.

Try it yourself

Find the one triplet whose sides add up to exactly 1000, a well-known Project Euler puzzle.

Edit the program above and press Run, or open it in the full editor to save it and share it. To watch it run line by line, open it in the visualizer.

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