(If you haven’t read the “Problem” post with the same title, go there first. This will make more sense if you do.)
We want to find all the Pythagorean triples (a,b,c). The first thing a mathematician would probably do is to try some small examples, gather some information, and then look for patterns within that information. For instance, if you only allow a to be a number from 1 to 10 and if you let b be between 1 and 50, here’s all the triples you get:
(3,4,5), (4,3,5), (5,12,13), (6,8,10), (7,24,25), (8,6,10), (8,15,17), (9,12,15), (9,40,41), (10,24,26).
Try to find some patterns in there. Look around for yourself…. the first thing you might notice is that some of them are really repeats – like (3,4,5) and (4,3,5) are really the same thing. So we can whittle down our list some without losing any information – in situations like this mathematicians usually choose the one where the first number is smaller, so I’ll do that, but it doesn’t really matter. Here’s the new list we got:
(3,4,5), (5,12,13), (6,8,10), (7,24,25), (8,15,17), (9,12,15), (9,40,41), (10,24,26).
Then, I look again for a pattern. There definitely looks like a lot of chaos, but there is at least one more thing we can pick out. Notice that some of them are just “multiples” of others. Like (3,4,5) can be made into (6,8,10) but doubling everything, and (9,12,15) is made by tripling everything. In fact, this motivates our first piece of knowledge:
Lemma: If (a,b,c) is a triple, then so is (na,nb,nc) for any positive whole number n.
Proof: We can see by simplifying that
So, the necessary equation is true, so (na,nb,nc) is a triple. So, we are done.
(Side note: Mathematicians use the words lemma, proposition, and theorem all to mean “a true statement.” The connotation of “lemma” is that this is a smaller “piece” that helps us get to some bigger, more important thing. A proposition and a theorem are the “bigger things”, and theorems are bigger and more important than propositions. They have nothing to do with “difficulty” per se, just how important they are to the questions we want to answer.)
This is a good step. What this means now is we can reduce our list even more, to the triples where the three numbers don’t have a common factor. These have a special name, called primitive triples. Now, we hit on a big math idea – that of building blocks. With a little bit of effort, the lemma we just found basically tells us that every Pythagorean triple is either primitive or is a multiple of a primitive triple. Therefore, if we can list all the primitive triples, we actually know all the triples. Going back to our list, we reduce down to the primitive triples…
(3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41).
As we have just seen, our original question “what are all the Pythagorean triples?” (which from now on will just be called triples) has been reduced to “what are all the primitive triples?” This turns out to be a question which can be addressed more directly.
To see the next pattern, we will now let a be larger than b again. So, we have a list
(3,4,5), (4,3,5), (5,12,13), (7,24,25), (8,15,17), (9,40,41)…
There is now another, not-so-obvious pattern here. Look at the a’s. We have a pattern 3,4,5,7,8,9,… 10 gets skipped, and if you keep going you see that 14 is also skipped. So 2, 6, 10, and 14 get skipped. That’s a pattern, or at least it looks like one. When you look at these numbers, this might give us an idea: if (a,b,c) is a primitive triple, neither a nor b can be a number like 4n+2. This turns out to be true:
Fact: If (a,b,c) is a primitive triple, then neither a nor b can be written in the form 4n+2 for n a whole number.
Proof: This time, I’m going to intentionally leave a few details out so that anyone who is curious can fill them in.
Now, where can we go from here? We’ve started looking at primitive triples. As a mathematician, this is good progress, but we are not done yet. I encourage the reader to think some about this, and in a second post on this question I will demonstrate a method of finding all of the primitive triples.