Observations on Pythagorean Triples

In a Pre-Calculus class several years back, to fill in (translation: kill) the last fifteen minutes of a class, I had my students search for as many different Pythagorean triples as they could find. While they were working, I examined the triples closely, like I never had before. I discovered a pattern which I had never noticed before. I use the word "discover" loosely, since I am sure that I am not the first person to uncover the patterns and characteristics to be described below.

Definitions and examples

A Pythagorean triple is a set of three strictly positive integers--a, b, and c--satisfying the equation a² + b² = c². One such Pythagorean triple is 3, 4, 5, since 3² + 4² = 5², or 9 + 16 = 25. Another such triple is 5, 12, 13, since 5² + 12² = 13². Check for yourself that this last equation is valid.

6, 8, 10 is also a Pythagorean triple, but I consider it to be part of the 3, 4, 5 family, since 6, 8, 10 are the same multiples of 3, 4, 5: 2x3=6, 2x4=8, 2x5=10. Other members of the 3, 4, 5 family would be, for example, 9, 12, 15 and 30, 40, 50. (Check that these are Pythagorean triples.) I'll term the 3, 4, 5 triple as the seed triple for its family.

Without loss of generality, let a be the least of the three numbers in the triple. There is a different family for each prime number a > 2. In other words, each prime number a > 2 generates a seed triple. The pattern I noticed was that the sum of the other two numbers of the seed triple, b and c, is equal to the square of a. In other words, a² = b + c. Again, this is only true when a > 2 is prime. I also noticed that b and c are consecutive integers.

With this information, all Pythagorean triples can be obtained [NEW]—or so I thought. J. Snyder of Anne Arundel Community College sent me an email, three years after my original “publication” on my website, with the following simple statement: 8, 15, 17. So, it was back to the drawing board. Fifteen minutes later, I emerged from the drawing board room (drawing room? board room?), covered with pencil shavings and eraser crumbs, with the realization that I had overlooked the lowly 2. Obviously I had!—read the first three paragraphs above. How many times did I write “a > 2”? My parents raised me better than to be prejudiced against any person or number, but it looks like I grew up as a Twoist. (Not a Taoist, mind you, although a Twoist is a Taoist with a Twist.) I must also consider each power of 2 as, potentially, the least number of a seed triple.

A closed form: a > 2 prime

For each a > 2 prime, there is a seed triple: .

Each seed triple generates a family of Pythagorean triples: where k is a positive integer.

[NEW] A closed form: powers of a = 2

For each positive integer n ≥ 3, there is a seed triple , which in turn generates its family , where k is a positive integer.

Note that for n = 2, we get the triple 3, 4, 5. In this case, is not the least number of the triple, but the formula is still valid. When n = 1, is not associated with any triple.

You might ask, “Well, what about the powers of primes a > 2?” Powers of primes are also multiples of primes.

Calculating your favorite Pythagorean triple

Suppose that you have an unhealthy psychological obsession with the number 11; you can determine the Pythagorean triple for a = 11. Note that 11 is a prime greater than 2, so use the observations two paragraphs up. Since a² = 121, you need to look for two consecutive integers whose sum is 121. There is only one such pair: 60 and 61. You can check that 11² + 60² = 61². Thus, the triple is 11, 60, 61. (There, I hope the demons have been put temporarily to rest.)

[NEW] Or start with the n = 6^th power of 2: We get the triple seed triple , which is 64, 1023, 1025. Check that it is really a Pythagorean triple.

Where's the (mathematical) beef?

It is easy to prove that the triple given by for a > 2 prime, or by for n ≥ 3, is a Pythagorean triple, but it would take some work to prove that all Pythagorean triples have one of these two forms. In the spirit of Fermat, I have furnished a link to the proof. Enjoy reading it, as I'm now off to play some basketball.

Final, nearly useless observations

For the seed triple a, b, c, where a >2 is prime, b is always even, and c, the hypotenal number, is always one greater than b and thus always odd. (Hypotenal is not a word, you say? It is in bobonics!)

For the seed triple a, b, c, where a = , n ≥ 3, b and c are both odd.

In both cases, the seed triple is composed of one even and two odd numbers.