Name: 42是新的33 - Numberphile (42 is the new 33 - Numberphile)
Uploaded: 2021-01-14T10:38:45.000Z
Duration: 13 min 40 s
Description: 【看影片學英語】數萬部 YouTube 影片，搭配英漢字典即點即查，輕鬆掌握單字發音與用法，長久累積看電影不必再看字幕。

I'm sorry that I didn't even notice I I learned from somebody else's posting that one of the numbers of prime, so I hadn't noticed that we found three injures Who's Cube?

Some 2 33 This number was immortalized in a paper by Bjorn Putin.

He pointed out that if you wanna solve that problem for the number 29 it's really easy.

If you move a bit up the number line and you ask about 30 we know a solution for that.

But the numbers are in the billions on that was only discovered in 1999 and Andrew's understand there are some numbers that you will never find one for.

So if your number, if you divide it by nine and you get a remainder of four or five that we know you can't do it.

So you made a brilliant video with Tim Browning on this att the time it was an unsolved problem.

There have been some attempts toe prove that this number isn't on the list, but those attempts failed and, well, I've just solved it.

I first saw the video, You know, around the time that it came out and I thought a bit about it, but didn't get very far.

It was actually the follow up video on 74.

I think it was made a couple of years ago, but I only saw it last month.

And that, um, spurred me to think more about it.

And I got hooked that this last number here 74 is actually representatives of some of three cues.

So about these numbers, I've got sand paper here.

So if you look at the end of the paper, he's put all of the numbers below 1000 for which we don't know the answer.

Brady, Do you notice anything about these numbers?

Now, if you stare at it a while, um, you might notice that their old divisible by three.

And if you take this a step further and divide them by nine.

Actually, this is a good plug for your casting out nine video.

Okay, so this one has remained or six when you divided by nine.

Another six starting to see a pattern here breaks with this one.

This is remainder three on so on and so forth.

So it turns out they all have remained or three or six when you divide them by nine.

So, uh, it's Tim explained in one of the of the follow up videos.

When you divide by nine, we know that if you get remainder four or five, then actually there's no solutions at all.

Um so it's the same kind of analysis shows that if you get remainder three or six, it's possible.

But only just it means the solutions for these numbers are likely to be more sports.

Okay, And that's what ends up, making them hard to find the modern conjectures.

Every one of these numbers they'll be infinitely many solutions.

Um, it's just they're extremely sparse on dso.

As we've seen, we want to solve an equation like X cubed plus y cubed plus Z cubed equals, say 33 on.

We're looking for solutions where all of the numbers are below some bounce.

Absolute value of why and absolutely Uzi are all below some bounds.

So imagine B is ah, 1,000,000 or something.

Now, on the face of it, this is a three dimensional problem.

Okay, we've got three variables to solve for X, y and Z.

And if you just went through all possibilities up to be that somewhere on the order of be cubed numbers to check, I said you can imagine B is a 1,000,000.

To put things in perspective, your smartphone could do a 1,000,000 things in the blink of an eye.

Okay, but if I say a 1,000,000 cubed all right, that's in the realm of possible but very expensive, um, to do on modern computers.

So it's not hard to see that in the search range.

There's loads of places where there's no point in looking Okay, So if I imagine that X, Y and Z are old, positive and large, and of course, there's no way that, uh, some of their cubes is gonna be a small number.

Okay, So already there's there's big parts of the search range that you that you can cut out.

If you take this idea to its logical conclusion, it's It's not hard to see how to get from be cubes down to be squared.

This was realized pretty early on even the very first computer investigations that people did in the fifties.

They had algorithms that were basically quadratic would work in this time.

Now that wasn't improved on for about 40 years.

And then a guy called No Mel Keys and Harvard.

Um, actually, in a newsgroup posting, um, he proposed another our rhythm, which is pretty amazing.

Actually, it manages to get down from B Square to basically be to the one power.

And it's basically what enables us to go to the range is that we are today, you know, tend to the 15 10 to the 16.

It turns out that l Keys algorithm is most efficient when you're looking for a solution for lots of numbers, um, a CZ we were for a while But you can see we're getting down to a pretty short list at this point.

So I started to think, Well, maybe there are other approaches that we could look at.

So what I'll do is I'll take this equation and then we'll move Z cubed over to the other side.

Okay, So that gives us X cubed, plus y cubed equals 33 minus Z cubed.

This is now X plus Y X squared minus x y.

Well, um, we have this factor that that we know is there.

And I promise we're getting to the end of elder breath.

Okay, so I'm gonna have on the left side just expert minus X y plus y squared right side 33 minus Z cubed over D.

Now it probably looks like I've made things a lot more complicated, right.

Now we before, but I want you to imagine for a moment that we know the values of Z and of D.

Or maybe we have some guests for Well, that means that everything in the right hand side is something I know or I can compute.

On the left hand side, I have a quadratic equation with two unknowns.

As everybody knows, if you got to variable to Seoul for you need two equations.

I've got one here, but there's another one that I used implicitly but didn't write down.

All right, so now I've got +22 equations to unknowns.

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42是新的33 - Numberphile (42 is the new 33 - Numberphile)

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