Pythagorean Triples

Introduction

From the Pythagorean theorem, we know the relation:

Three positive integers $(a,b,c)$ that satisfy this equation are called Pythagorean triples. If the three numbers have no common factor other than 1, the triple is called a primitive Pythagorean triple.

Basic Formula

If $m$ and $n$ are positive integers with $m>n>0$, then one standard form of a Pythagorean triple is:

Proof

Let:

Then:

So $(2mn,\ m^2-n^2,\ m^2+n^2)$ really is a Pythagorean triple.

Area and Perimeter

If the legs and hypotenuse of a right triangle are written in this form, then:

Area

Perimeter

In some cases, the area and the perimeter are equal. Starting from:

we can derive:

Examples:

• if $m=3$ and $n=1$, then area = perimeter = 24
• if $m=3$ and $n=2$, then area = perimeter = 30

Types of Pythagorean Triples

1. Primitive Pythagorean Triples

A Pythagorean triple $(a,b,c)$ is called primitive if:

Some useful properties:

• at least one of $a$, $b$, or $c$ is odd
• one of $a$ or $b$ is even
• $c$ is always odd

2. Twin Primitive Pythagorean Triples

A primitive triple is called twin primitive when the difference between the hypotenuse and the longer leg is 1.

Using:

if $a>b$, the twin condition gives:

So:

If $b>a$, then:

which has no positive integer solution.

3. Composite Pythagorean Triples

A triple is called composite when the three numbers still have a common factor greater than 1.

Example 1

Prove that if one of $a$ or $b$ is even in a primitive Pythagorean triple $(a,b,c)$, then $c$ must be odd.

Solution

Use an indirect proof. Suppose $c$ is even while $a$ and $b$ are odd.

Write:

Substitute into the Pythagorean equation:

The left-hand side is not an integer, but $n^2$ must be an integer. This contradiction shows that $c$ cannot be even. Therefore, $c$ must be odd.

Example 2

Find all primitive Pythagorean triples $(a,b,c)$ with $b$ even and $c \le 40$.

Solution

Take:

with conditions:

• $m>n>0$
• $m$ and $n$ are relatively prime
• $m$ and $n$ are not both odd
• $m^2+n^2 \le 40$

The valid results are:

How to Generate Pythagorean Triples

General steps:

1. choose two natural numbers $m$ and $n$ with $m>n$
2. compute $m^2-n^2$, $2mn$, and $m^2+n^2$
3. those three values form a Pythagorean triple

Examples:

Notes:

• every multiple of a Pythagorean triple is also a Pythagorean triple
• Pythagorean triples are not limited to integers; they also appear with rational or irrational numbers, such as $(3,3,3\sqrt{2})$ or $\left(\frac{3}{2},2,\frac{5}{2}\right)$

Closing

Pythagorean triples provide a fast way to build right triangles without starting from scratch every time. Once the basic formula and the primitive conditions are understood, many useful numerical patterns become easy to generate.