Introduction

One of the most beautiful results in elementary mathematics is the Pythagorean Theorem, which states:

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Mathematically,
[
a^2 + b^2 = c^2
]

Whenever three positive integers ((a, b, c)) satisfy this equation, they form what is called a Pythagorean Triplet. These triplets have fascinated mathematicians for over 2500 years and appear in geometry, number theory, Olympiads, and real-world applications.

What Is a Pythagorean Triplet?

A Pythagorean triplet is a set of three natural numbers ((a, b, c)) such that:
[
a^2 + b^2 = c^2
]
where:

• (a) and (b) are the legs of a right triangle
• (c) is the hypotenuse

Examples

• ((3, 4, 5)) → (3^2 + 4^2 = 9 + 16 = 25 = 5^2)
• ((5, 12, 13))
• ((8, 15, 17))

Primitive and Non‑Primitive Triplets

Primitive Pythagorean Triplets

A triplet ((a, b, c)) is primitive if:
[
\gcd(a, b, c) = 1
]

That is, the three numbers have no common factor greater than 1.

Examples:

• ((3, 4, 5))
• ((5, 12, 13))
• ((7, 24, 25))

Non‑Primitive Pythagorean Triplets

If a triplet is obtained by multiplying a primitive triplet by a constant (k > 1), it is non‑primitive.

Examples:

• ((6, 8, 10) = 2(3, 4, 5))
• ((9, 12, 15) = 3(3, 4, 5))

Generating Pythagorean Triplets (Euclid’s Formula)

All primitive Pythagorean triplets can be generated using Euclid’s Formula.

Let (m > n) be two positive integers such that:

• (m) and (n) are coprime
• One of them is even

Then:
[
a = m^2 – n^2, \quad b = 2mn, \quad c = m^2 + n^2
]

Example

Take (m = 2, n = 1):
[
a = 3, \quad b = 4, \quad c = 5
]

Take (m = 3, n = 2):
[
a = 5, \quad b = 12, \quad c = 13
]

This formula is complete: every primitive Pythagorean triplet arises from it.

Key Properties of Pythagorean Triplets

1. In every primitive triplet:
   • One of (a, b) is even
   • The other is odd
   • (c) is always odd
2. The smallest triplet is ((3, 4, 5))
3. No primitive triplet has both (a) and (b) even or both odd
4. There are infinitely many Pythagorean triplets

Special Patterns and Families

Consecutive Numbers

Some triplets involve consecutive integers:
[
(3, 4, 5), \quad (20, 21, 29)
]

Triplets with Hypotenuse One More Than a Leg

These patterns are frequently used in Olympiad problems.

Pythagorean Triplets in Geometry

Pythagorean triplets represent exact integer‑length right triangles, which are useful in:

• Coordinate geometry
• Construction problems
• Lattice geometry

Example:
A triangle with vertices ((0,0), (3,0), (0,4)) is right‑angled with hypotenuse 5.

Applications in Real Life

• Architecture & Construction: Ensuring right angles
• Navigation & GPS: Distance calculations
• Computer Graphics: Pixel geometry
• Physics & Engineering: Vector magnitudes

Ancient civilizations like the Babylonians and Egyptians used triplets long before modern algebra.

Olympiad and Competitive Exam Importance

Pythagorean triplets frequently appear in:

• Olympiad geometry
• Number theory problems
• JEE Advanced
• SAT, ACT, IB, AP Mathematics

They are often hidden inside algebraic identities, coordinate problems, or Diophantine equations.

Common Mistakes Students Make

• Assuming all solutions are multiples of ((3,4,5))
• Forgetting coprimality conditions
• Ignoring parity (odd/even structure)
• Missing Euclid’s formula in disguised problems

Practice Problems

1. Find all Pythagorean triplets with (c = 25).
2. Prove that ((2n, n^2 – 1, n^2 + 1)) forms a triplet for odd (n).
3. Can ((10, 24, 26)) be primitive? Why or why not?

Final Thoughts

Pythagorean triplets are not just a list of numbers—they reveal the deep structure of integers, geometry, and algebra working together. Mastering them builds intuition that pays dividends across mathematics, from school exams to Olympiad‑level problem solving.

If you truly understand Pythagorean triplets, you don’t just know a theorem—you understand a language of right‑angled geometry.

Written by Rishabh Kumar
Founder, Mathematics Elevate Academy