Baudhāyana-Pythagoras triples, and how to make infinitely many

0:00Three, four, five. Nine plus sixteen is twenty-five, so those three whole numbers are the sides of a right triangle. Three whole numbers that do that are worth a name, and they have one: a triple. Both words matter. The relation has to hold, and all three have to be whole numbers. Halve them and you get one point five, two, two point five. That still satisfies the relation exactly.

0:31But it is not a triple, because two of the three are no longer whole numbers. It is the same triangle, drawn at half the size. What was lost was not the shape. It was the whole numbers. A side of no length, or a negative one, satisfies it too, and is no triangle at all. Here is a list of six of them, written down long ago. Three, four, five. Five, twelve, thirteen. Eight, fifteen, seventeen. Seven, twenty-four, twenty-five. Twelve, thirty-five, thirty-seven. And fifteen, thirty-six, thirty-nine.

1:12Every one of the six checks out, which is worth doing rather than assuming. But one of the six is not like the others. Fifteen, thirty-six, thirty-nine. Divide all three by three and you get five, twelve, thirteen, which is already on the list. So it is not a new triangle. It is one of the others, drawn three times bigger. Six triples. Five triangles. Hold on to that gap, because the whole video is about it.

1:45Now let us take nobody's word for it, and find every triple whose three numbers are twenty or less. Not remember them. Search for them, one combination at a time. There are six. Three, four, five. Five, twelve, thirteen. Six, eight, ten. Eight, fifteen, seventeen. Nine, twelve, fifteen. Twelve, sixteen, twenty. And look at what turns up. Six, eight, ten is three, four, five doubled. Nine, twelve, fifteen is it tripled. Twelve, sixteen, twenty is it quadrupled.

2:24So six triples again, and this time only three triangles. Half of what we found was one triangle in different sizes. That pattern is a machine worth turning the handle on. Take three, four, five and multiply every term by ten. Thirty, forty, fifty. Nine hundred plus sixteen hundred is two thousand five hundred. It works. Multiply by a hundred. Three hundred, four hundred, five hundred. It works again. At this point most of us would say: it always works. And most of us would be right.

3:04But saying it always works, on the evidence of two cases, is a guess with good manners. That has a name: a conjecture. Turning a conjecture into a fact takes something the two examples cannot supply. The wrong way to settle it is to try more numbers. Try a thousand. Try a million. You will be right every time and you will never be finished. The right way is to stop using numbers.

3:35Take three k, four k, five k, where k stands for any whole number at all. Three k squared is nine k squared. Four k squared is sixteen k squared. Add them: twenty-five k squared. And five k squared is also twenty-five k squared. The two sides are the same expression. That is not evidence any more. That is settled, for every k at once, including the ones nobody will ever write down.

4:09And notice that three, four and five did no work in that argument. Run it again with letters where the numbers were. Take any triple a, b, c. Scale all three by k. k a squared is k squared times a squared. k b squared is k squared times b squared. Add them and pull the k squared out in front: k squared, times a squared plus b squared. But a squared plus b squared is c squared, because a, b, c was a triple to begin with. So the total is k squared c squared, which is k c squared.

4:50One argument. Every triple, every whole number. There are infinitely many triples, and we have just proved it. So that question is answered, except it was not the question anyone was really asking. Take three, four, five and scale it by one, two, three, four, five and six. Six triples. Every one of them different. Every one of them genuinely a triple. Now divide each one back down by whatever its terms share. Every single one collapses to three, four, five.

5:27Six triples. One triangle. The scaling machine makes infinitely many triples and never once a new shape. It cannot: multiplying every side by the same number is exactly what leaving the angles alone means. So the interesting question is still standing, and it is: how many different right triangles have whole-number sides? To ask that properly we need a word for a triple that is not a magnified copy of anything.

6:01It is one whose three terms share no factor above one. Those are the primitive ones. Three, four, five shares nothing. Nine, twelve, fifteen shares three, so it is a copy. Careful here: primitive does not mean small. Eight, fifteen, seventeen is primitive, and it is bigger than nine, twelve, fifteen, which is not. And every triple is one or the other. Divide any triple by whatever its three terms share, and what is left is primitive, and still a triple.

6:38Nine, twelve, fifteen divided by three is three, four, five. The scaling proof run backwards. So finding all the primitive ones would be finding all of them. That is the real question, and scaling has not touched it. The answer comes from somewhere that looks like it has nothing to do with triangles. Add up the odd numbers, starting from one, and see what you land on. One is one. One plus three is four. One plus three plus five is nine. Add seven as well and you get sixteen.

7:16One, four, nine, sixteen. Every one of them a square. Add the first n odd numbers and you get n squared, every time. Now write it in two halves. The first n minus one already add to n minus one, squared. So one more odd number takes you from one square to the next. n minus one, squared, plus the nth odd number, equals n squared. Look at that sentence again with a triangle in mind.

7:49Something squared, plus something, equals something else squared. It is nearly the relation we want. The only thing missing is that the middle term be a square too. So here is the whole machine: find an odd number that is also a square. Nine is odd, and nine is three squared. Which odd number is it? Nine is two times five, minus one, so it is the fifth. So the first five odd numbers add to five squared, and the first four add to four squared. Four squared plus nine is five squared.

8:27And nine is three squared. Three, four, five falls straight out. Twenty-five is odd, and it is five squared. It is the thirteenth odd number, because twenty-five is two times thirteen minus one. So twelve squared plus twenty-five is thirteen squared. Five, twelve, thirteen. Feed it forty-nine and out comes seven, twenty-four, twenty-five. Eighty-one gives nine, forty, forty-one. A hundred and twenty-one gives eleven, sixty, sixty-one. A hundred and sixty-nine gives thirteen, eighty-four, eighty-five. Two hundred and twenty-five gives fifteen, one hundred and twelve, one hundred and thirteen.

9:16Seven odd squares, seven triples, and this time seven different triangles. Not one is a copy of another. And every one of them is primitive, for a reason you can see in the numbers: the longer leg and the longest side are always one apart. Anything dividing all three would have to divide the difference between two of them, and that difference is one. Nothing above one divides one. One warning, though. Do not try this on every odd number.

9:51Seven, eleven, fifteen are all odd, and none of them is a square, so none of them gives you anything. Between three and twenty-five, only two odd numbers work out of twelve. And the machine has a blind spot of its own. Every triple it makes has that gap of one. So it can never produce eight, fifteen, seventeen, where the gap is two. Nor twelve, thirty-five, thirty-seven. Nor twenty, twenty-one, twenty-nine, where the gap is eight.

10:24Two of those three were on the list we started with, so that list was never one machine's output. So: infinitely many triples, twice over, for two completely different reasons. One machine copies what you have, forever. The other keeps finding shapes nobody had. And there are still triangles neither of them reaches. The counting is not finished, and that is the good news.