Pythagorean Triples Generator
Generate primitive Pythagorean triples (a, b, c) with a² + b² = c².
Find triples up to a chosen hypotenuse and check whether three integers form a triple.
Pythagorean Triples
A Pythagorean triple is a set of three positive integers (a, b, c) such that:
a² + b² = c²
These are integer-sided right triangles. The most familiar example is (3, 4, 5): 9 + 16 = 25.
Primitive vs Non-Primitive
A triple is primitive if gcd(a, b, c) = 1 — that is, the three values share no common factor. Every triple is a primitive triple multiplied by some positive integer:
(3, 4, 5) → (6, 8, 10), (9, 12, 15), (12, 16, 20), …
Euclid’s Formula: Generating All Primitive Triples
For coprime integers m > n > 0 with one even and one odd:
a = m² − n² b = 2 × m × n c = m² + n²
This produces every primitive Pythagorean triple exactly once.
| (m, n) | (a, b, c) |
|---|---|
| (2, 1) | (3, 4, 5) |
| (3, 2) | (5, 12, 13) |
| (4, 1) | (15, 8, 17) |
| (4, 3) | (7, 24, 25) |
| (5, 2) | (21, 20, 29) |
| (5, 4) | (9, 40, 41) |
| (6, 1) | (35, 12, 37) |
| (6, 5) | (11, 60, 61) |
| (7, 2) | (45, 28, 53) |
Famous Triples
The earliest evidence for Pythagorean triples comes from Babylonian tablet Plimpton 322 (c. 1800 BCE), which lists 15 triples — over 1000 years before Pythagoras. The modern proof of Euclid’s formula appears in Elements, Book X.
Properties
| Property | Statement |
|---|---|
| At least one of a, b is divisible by 3 | Always true |
| At least one of a, b is divisible by 4 | Always true |
| At least one of a, b, c is divisible by 5 | Always true |
| Exactly one of a, b is odd | True for primitives |
| c is always odd | True for primitives |
| Inradius | r = (a + b − c) / 2, always an integer |
Worked Example: Verify (20, 21, 29)
20² + 21² = 400 + 441 = 841 = 29² ✓ This is a primitive triple from m = 5, n = 2:
- a = 25 − 4 = 21
- b = 2 × 5 × 2 = 20
- c = 25 + 4 = 29
Modern Uses
| Field | Use |
|---|---|
| Construction | Right-angle layout with measuring rope |
| Surveying | Squaring large rectangular plots |
| Cryptography | Underlying number-theory exercises |
| Computer graphics | Discrete pixel right triangles |
| Recreational math | Fermat’s Last Theorem (its negative cousin) |
Connection to Fermat’s Last Theorem
Fermat’s Last Theorem states there are no triples of positive integers (a, b, c) with aⁿ + bⁿ = cⁿ for n ≥ 3. For n = 2 the count is infinite — that is the entire content of this calculator. For n ≥ 3, Andrew Wiles proved in 1994 that none exist.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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