Last week I posed a challenge that came from my high school computer teacher: Can you find an efficient method (or “algorithm”) to generate Pythagorean triplets?

As a reminder, a Pythagorean triplet is a set of three integers that satisfy the Pythagorean equation:

This equation is an example of a Diophantine equation. Diophantine equations are just polynomials that require the solutions to be integers. Because of that, they can be hard to solve.

**The Brute Force Method**

The first method most people try is the brute force method: they simply try a bunch of triplets and see if they work. Not exactly elegant, and very, very slow. But it does work.

**A Substitution Method**

I didn’t like the idea of having to try triplets over and over again to see if they worked. I wanted a solution you could simply calculate. Here’s the approach I came up with:

- First, we need to find some equations we can use to calculate , , and .
- Second, those equations, when plugged into the Pythagorean equation, must give us a true equation.
- Finally, we can then use those equations to calculate values forÂ , , and which we know will satisfy the Pythagorean equation, and therefor will be Pythagorean triplets.

The equations I created used the variablesÂ , and .

If we plug those into the Pythagorean equation, we get:

the left side of that equation becomes:

and the right side becomes:

Since both sides become the same thing, we know the equation is true. That’s great news, because it means our equations to calculateÂ , , and do in fact work in the Pythagorean equation.

So, how do we use them to find Pythagorean triplets?

Well, we know we never want

.

Because of that, we have to constrain our choices to

.

So now we just choose all of the positive integers and plug them into the equations. Â Here’s what we get:

and so on.

As you can imagine, this is far, far faster for generating Pythagorean triplets than the standard “Brute Force” try-every-possible-combination approach.

Let me know if you come up with a better approach.