Trippy Triples

Have you ever heard of a 3-4-5 triangle? It is a right-angled triangle with sides of length 3 and 4 and 5. It looks like this:

The neat thing about this triangle is that the sides are all whole numbers. This is not always the case for right-angled triangles. For example, a triangle with sides of length 2 and 3 has a hypotenuse (the side across from the right angle) of length 3.6.

A set of whole numbers like (3, 4, 5) that create a right-angled triangle is called a Pythagorean triple. How many triples are there? Well, a triple has to be a right-angled triangle. One way to check if you can construct a right-angled triangle from 3 numbers is to plug the numbers into the Pythagorean Theorem.

$a^2+b^2=c^2$

$3^2+4^2=5^2$

$9+16=25$

$25=25$

Since both sides are equal, (3, 4, 5) is a triple. However, consider (4, 5, 6):

$a^2+b^2=c^2$

$4^2+5^2=6^2$

$16+25=36$

$41 \neq 36$

Since both sides of the equation are not equal, (4, 5, 6) is not a triple. Or visually, you can see how it is impossible to make a right-triangle with those side lengths:

Ok, so we know that (3, 4, 5) is a triple, but how can we find more? A bit of investigation reveals that you can multiply each number of a triple by two and this new set will also be a triple (6, 8, 10).

$6^2+8^2=10^2$

$36+64=100$

What if we multiply by 3? Will (9, 12, 15) be a triple? You bet!

$9^2+12^2=15^2$

This pattern continues. In general, once you have found a triple, you can generate infinitely many new triples. Pretty cool. But also not cool. Because, although these new triples are new, they are actually just scaled up versions of the original triple. How could we make a different kind of triple? There just so happens to be a formula for generating new triples1. What we find when we use the formula, is that each prime number (3, 5, 7, 11, 13, etc.) generates a unique triple. For example, (5, 12, 13):

$5^2+12^2=13^2$

$25+144=169$

Or (7, 24, 25):

$7^2+24^2=25^2$

$49+579=625$

Each of these new triples can be scaled up to create an infinite number of triples. However, there are also an infinite number of prime numbers. This means that there are an infinite number of triples that start with a prime, and each of these triples generates an infinite number of scaled up triples. Trippy triples.

1For those of you who want to see how the formula works, here is a behind the scenes look at Pythagorean triples.

We start with the following set of three equations which relates the triples to 2 unknown variables m and n:

$A=m^2-n^2$

$B = 2mn$

$C = m^2 + n^2$

Consider $A = m^2 - n^2$. We can factor the RHS to get $A = (m-n)(m+n)$. However, A is prime, so the only factors of A are 1 and A. Thus:

$m-n = 1$

$m+n = A$

Solving the above system of equations for m and n yields the following:

$m = \dfrac{A + 1}{2}$

$n = \dfrac{A - 1}{2}$

In this form, m and n are completely determined by A. Thus, B and C are also completely determined by A.

$B = 2 * \dfrac{A+1}{2} * \dfrac{A-1}{2} = \dfrac{A^2 - 1}{ 2}$

$C = \dfrac{(A+1)^2}{2} + \dfrac{(A-1)^2}{2} = \dfrac{2A^2 + 2}{4} = \dfrac{A^2+1}{2}$

This tells us how to find B and C when given any prime A. It also tells us that B and C will be consecutive integers. How interesting.