Talia: Hey, Mom. Can you help me with my math homework?Talia gave me her paper, and I read the definition of a Pythagorean Triple: "Any set of three whole numbers that satisfies the Pythagorean theorem. Examples include (5, 12, 13) and (7, 24, 25)."
Me: Sure, what is it?
Talia: We need to figure out some Pythagorean Triples.
Me: I know who Pythagoras was and what the Pythagorean Theorem is, but let me see your paper. I don't know what the triple means.
For those of you who don't remember, the Pythagorean Theorem states: "That the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
(BTW I found that little definition on Wikipedia, but you can find it other places, too.)
In short, the girls needed to find three numbers that satisfied the theorem. And their teacher gave them a hint. Keep the numbers under ten, he said.
Me: This won't be too hard. Let's figure it out.So I prodded Talia and she made a chart where she chose a c and then an a and b and then calculated the square to see if the variables worked. It didn't take long for her to find that (3, 4, 5) and (6, 8, 10) were the answers her teacher probably wanted. I helped Zoe do the same thing once Talia was done.
Me: Well, your dad probably has some nifty equation to do it, but I think you should just use good old trial and error.
As soon as Tim walked back into the house, I asked him about the homework.
Me: I'm sure there's another way to do it, but I just had the kids choose numbers and test them.Tim walked over to his work bag, extracted a notebook, and opened to a page, somewhere in the middle. This is what it said on his paper:
Tim: Just a moment.
Theorem: There exists infinitely many Pythagorean triples.I don't need to write the rest. I smirked when I saw it. Of course there was an equation that could figure out the same thing we had by testing numbers. And of course, Tim had it, at hand, proof written out. In his notebook. Something he'd done it for fun, some other time in his life.
Proof: Let p be an odd integer > 1.
Corresponds to a right triangle with legs of length...
I tease, but to be honest, I'm also jealous.