3 * 4 = 2 * 6. You can multiply different bunches of numbers and get the same result. But you can't do this if the numbers you multiply are all prime. Why is that?
A basic fact about numbers, yet almost nobody graduates from 12+ years of compulsory math education able to explain it. Math class is a waste of time, fixated on making you a human calculator, John Henry against the electronics industry, memorizing tables and running algorithms.
Math class is boring and everyone is right to be bored by it. You have a brain to do better things with.
A closely related question, that takes a little longer to ask, is this:
Consider a fraction like 36/60. We can scale this fraction down, because 36 and 60 are both divisible by 4. Scaling numerator and denominator down by 4, we get 9/15. We can also scale 36/60 down using that 36 and 60 are both divisible by 6. Scaling down by 6, 36/60 becomes 6/10.
But neither of these was scaled down as far as possible. 9/15 can still be scaled down further, to 3/5, which can't be scaled down anymore. 6/10 can also be scaled down to 3/5. No matter what starting choice we made, if we keep going, we end up at 3/5.
Why did it work out this way, though? Why couldn't it have been that 6/10 scales down to one thing, and 9/15 scales down to a different thing? Why is it that no matter what choices we make along the way, we always end at the same place, when scaling down a fraction?
You were told these things so often without explanation that you took them for granted. You stopped seeing there even was a mystery here someone could ask about. And indeed, there is a simple explanation. But can you explain it? It's no use just saying "That's how it is" by fiat.
You don't have to be interested in any of these mystery puzzles, these opportunities for stories and insight, but there's at least something there it is possible to be interested in. There is nothing worthy of interest in drilling calculating 50 "this times that"s by hand a day.
That's my actual point, that everything interesting in math has been systematically hidden from you, and what you've been made to waste your time on instead sucks shit. There is no point training a human to be a dumb machine, but that's what they do.
It's only natural to end up thinking that's what math is. It'd be better if you'd been allowed that time to pursue your actual interests, whatever they be, inside or outside math. But you are denied freedom to decide how to spend the limited time of your own life from early on.
Alright, let's give an explanation for the fraction thing, and then I'll use it to return to the prime thing.
I'll present the explanation in a visual way, since people often like that, but the logic of this explanation could just as well be stated in non-geometric terms.
I'll present the explanation in a visual way, since people often like that, but the logic of this explanation could just as well be stated in non-geometric terms.
Consider a square grid, and draw some other line on top of it, starting from some grid point. Any line you like. 
Does this line run through any other grid points? Maybe, maybe not, it depends on the line you chose.
In this particular case, I chose the line to also go through another point, far off, 36 units to the right and 60 units up. But I'm interested in what it's like near the Start.
In this particular case, I chose the line to also go through another point, far off, 36 units to the right and 60 units up. But I'm interested in what it's like near the Start.
This line goes through other grid points too, much closer to the Start. Indeed, the closest grid point it goes through is only 3 units over and 5 units up.
1 unit over, it doesn't go through a grid point. 2 units over, it doesn't go through a grid point. But 3 over, it does.
1 unit over, it doesn't go through a grid point. 2 units over, it doesn't go through a grid point. But 3 over, it does.
What's the NEXT grid point the line goes through? Well, every grid point is exactly like every other grid point. Moving out along this line's slope required us to go 3 over and 5 up from the Start before the next closest grid point, but there's nothing special about Start.
So from ANY grid point this line goes through, we have to go 3 over and 5 up before the next one. 1 over won't work, 2 over won't work, but 3 over will.
Moving over another 3 and up another 5, we end up 6 over and 10 up from the Start.
Moving over another 3 and up another 5, we end up 6 over and 10 up from the Start.
And the next grid point after that? Same thing again. Each one is just like the Start. Each time the line hits a grid point, it has to go over 3 more and up 5 more to hit a next grid point.
(3, 5), (6, 10), (9, 15), (12, 20), …, these are the grid points the line goes through.
(3, 5), (6, 10), (9, 15), (12, 20), …, these are the grid points the line goes through.
In other words, the line goes through every multiple of (3, 5), and nothing else.
It'll eventually get to the (36, 60) we constructed this line to go through, and thus (36, 60) itself is a multiple of (3, 5). Every grid point on the line is a multiple of the smallest one.
It'll eventually get to the (36, 60) we constructed this line to go through, and thus (36, 60) itself is a multiple of (3, 5). Every grid point on the line is a multiple of the smallest one.
There was nothing special about this line. This story works the same for any line. If it hits grid points at all, it hits them at regular intervals. It will hit one closest to the Start, and then all the other ones it hits will be the multiples of this first one.
I've been showing number-pairs as points, but a number-pair can also be called a fraction. The grid points (A, B) on a line are the fractions A/B which are quantitatively equal to the fraction of a unit the line moves over by when we go up 1 unit.
And since all the grid points on a line are multiples of the smallest one, we conclude that every fraction has to be a "multiple" (i.e., scaling up by a whole number) of the lowest denominator fraction it's quantitatively equal to.
This is why, no matter how you make choices along the way in cancelling common divisors from numerator and denominator to reduce a fraction, you always wind up at the same smallest numerator and denominator in the end.
You've used this fact a million times in class, you've internalized it as obvious by rote, but it's not quite so tautological. There is in fact a question to ask and story to tell, explaining why it works that way.
(It's a pithy story, but unfortunately looks large here, because Twitter makes every paragraph look like a novel.)
Now back to primes. The most basic fact about primes you are told by rote, but never given an explanation for, is this: If two numbers aren't divisible by a particular prime, then neither is their product.
(This is called "Euclid's lemma", but names don't matter)
(This is called "Euclid's lemma", but names don't matter)
Why is this? Suppose P is prime, and A x B = P x something. In other words, A / P = something / B. As we just saw, whenever fractions are equal, they're both "multiples" of the lowest-terms version of that fraction.
But since P is prime, the only things it's a multiple of are P and 1. So either A / P is already in lowest terms and the other denominator B is a multiple of P. Or the lowest-terms version has a denominator of 1, meaning A / P is a whole number, meaning A is a multiple of P.
So if neither A nor B are multiples of prime P, then A x B can't be a multiple of P either.
This was not an obvious tautology, primitive and unquestionable. It's a fact with an explanation, a story in answer to questioning it. It gets explained through a story like the above.
This was not an obvious tautology, primitive and unquestionable. It's a fact with an explanation, a story in answer to questioning it. It gets explained through a story like the above.
From this, the uniqueness of prime factorization is straightforward.
If you multiply together a bunch of primes that aren't P, then the result won't be divisible by P. The only way to get a result divisible by P is for one of the primes to be P, by what we just showed.
If you multiply together a bunch of primes that aren't P, then the result won't be divisible by P. The only way to get a result divisible by P is for one of the primes to be P, by what we just showed.
And indeed, if we use P as a factor three times and then a bunch of other primes, we get something divisible by P three times, but no more. The only way to multiply a bunch of primes to get X is to include each prime P precisely as many times as X is divisible by P.
That's why prime factorization is unique.
The End.
The End.
I know, no one reads Twitter threads of this sort, it's not the right medium, I should never have started this dumb thing, ill-advised posting, I never learn. But, having started it, I had to pay off the explanations I promised.
I chose an explanation path that didn't go through the fact that, given any bunch of numbers (e.g., 6 and 15), you can always add and subtract them to get something that divides all of them (e.g., 15 - 6 - 6 = 3, and 3 divides both 6 and 15).
That is, I didn't go through "Bézout's Lemma" explicitly, although it's there implicitly, through a glass darkly.
It's much more fruitful and clean in many ways to explain that. It's also a concrete result that people don't usually find already obvious.
But it's a little more abstract feeling than thinking of lines and grid points, so I went with the latter for this.
The End, for real.
But it's a little more abstract feeling than thinking of lines and grid points, so I went with the latter for this.
The End, for real.
Anyway, the explanation is here:
I decided I liked a different explanation of Euclid's lemma much better. You can find it here.
I decided I liked a different explanation of Euclid's lemma much better. You can find it here.
