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Pythagorean Quadruples

Astute geometry students know that when a problem involves a 3 and a 4, the answer will involve a 5 and the solution will make use of a right triangle somehow.

The triple of numbers $(3,4,5)$ is called a Pythagorean triple because they are integers that form the side lengths of a right triangle, which we know because of the Pythagorean Theorem:

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

(See this other post on Pythagorean triples.) Teachers love to make use of Pythagorean triples to keep numbers nice in geometric problems.

Less well-known are Pythagorean quadruples, groups of integers $(a,b,c,d)$ such that $a^2+b^2+c^2=d^2$. These quadruples allow teachers to ask 3-dimensional geometry questions with nice numbers. For example, if a bird flew 6 miles north, then turned and flew 3 miles west, and then turned and flew 2 miles straight up, it would end up 7 miles from where it started.

The two most common Pythagorean quadruples are the smallest ones, $(1,2,2,3)$ and $(2,3,6,7)$. As a teacher, I never knew any other Pythagorean quadruples, probably because my classes didn’t do enough 3-dimensional geometry to require them. But it turns out there are a lot more, more than I expected. I wrote a computer program to output all of these triples with each of $a$, $b$, and $c$ being 20 or lower.

Looking at this table, it appears that there are several interesting patterns in these quadruples.

  1. Consider the quadruples $(1,2,2,3)$, $(2,3,6,7)$, $(3,4,12,13)$, and $(4,5,20,21)$. Each of these quadruples starts with consecutive integers. Moreover, the third number is the product of the first two, and the fourth is one more than that. In symbols, it seems that $\Big(a,a+1, a\cdot(a+1),a\cdot(a+1)+1\Big)$ is always a Pythagorean quadruple.
  2. We can find Pythagorean quadruples with two 2s, two 4s, two 6s, two 12s, two 20s, etc. All of these numbers are even.
    • Is it possible to find Pythagorean quadruples with two copies of some odd number?
    • Can we find a Pythagorean quadruples with two copies of any even number, or are there some even numbers that get left out?

I’ll try to answer these questions in a future post. Do you see any other fascinating patterns we should look into?

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