Did the AI brute-force it?

1:26:09actually my next point. Okay. >> We can actually we can actually ask did Fable just brute force. >> Right. Right. Which I >> didn't just do like supercomputer nonsense >> cuz they have a lot of computers. >> Yeah. And and there's history behind supercomputer nonsense. Okay. So let's talk about like this is a counter example. Okay. And historically computation has been very good at producing counter examples. Okay. way back in the 1960s actually. So let's talk about um Oilers's conjecture. Oilers's conjecture is the following that um you remember um we were talking about like from theorem and the Pythagorean theorem like the Pythagorean theorem is that like the the sum of two squares can equal another square. So like two >> two thingies to the power of two equals another thing to the power of two.

1:26:51Trivial example 32 + 42 = 52 because 9 + 16 = 25. Um Plato way back in the day had found another kind of Pythagorean looking thing where he showed that 3 cub + 4 cub + 5 cubed is actually 6 cubed >> which is kind of cool. >> It's called uh it's called Plato's number. >> Mhm. >> Um so all right the Oiler looked at this looked at Plato's number and he said well um look if if if I need in that case I needed three cubes >> Yes. >> to get to another cube. You're right. >> Okay. So, Oiler is like, what if I need n >> to the nth power to get to another nth

1:27:32power? I need at least n things to the nth power to get to another nth power. So, if I want to if I want something like a 4th plus b 4th plus c 4th, I would need another d 4th in order to get to an e to the 4th. >> He was trying to generalize number. >> Exactly. He's trying to general cuz pyagorean theorem already happened, right? 32 + 42= 52. Plato's number is saying okay I need three cubes to get to another cube so maybe I need four to the fourth power to get to a fourth I need five to the fifth power right and that's that's Oiler's conjecture remains untrue for a very long time lo and behold um Lander and Parkin in a very very famous mathematical paper one of the shortest

1:28:14math papers of all [laughter] time right they're like uh direct search with the CDC 6000 66000 which is one of the first successful supercomputers they just brute forced it. >> Y >> and they found a counter example. >> Um 27 to the 5ifth power plus 84 to the 5th power plus 110 to the 5ifth power plus 133 to the 5ifth power equals 144 to the fifth power. So I only needed four fifth power thingies to get to a fifth power. >> We don't like that. >> Yeah. And they were just like it's the smallest instance >> in which five fifth powers. There you go. >> Sorry. >> Sorry Oiler. You're wrong. crazy to say the oiler is [laughter] wrong,

1:28:56>> but but this this is a perfect point. This is why people immediately go to again, you know, the >> oh, they just brute forced it. >> Yeah. Right. Right. And this in this case, the brute force it could work because the sample space that they were they they were just like, okay, let's just go for fifth powers. Let's go all the way up to like 200 for all the numbers and let's just try it out. >> And it happened to be there. And the search space is small enough where with a big enough supercomput I can do it. >> And this was some some time ago. >> Yeah, this was in the '60s. And now we found more and more exceptions of Oilers's conjecture. It's rare. It's increasingly rare. And there's a lot of mathematics about how rare it is. And that's its own, you know, mathematical field. But let's consider could Fable

1:29:39have just like brute forced it, >> right? >> I don't think so. I think this is fundamentally different, right? The search space is actually I think too large because here what we'd have to do is test every single function >> in of polomials right there's x y and z. So let's just do like okay it found like the seven degree polinomial right there was an x to the seven there um >> if you just think about seven degree polomials and seven degree maps so you can have seven 7° in x 7° in y 7° in z right and each of those can have different numbers >> right so that's first of all that's 343

1:30:19different polomial terms each of those polomial terms can then make up 1300 cubic equations And then the coefficients of that equation, right? Is it 1 x^2 y? Is it 2x^2 y? Is it 1 x cubed y? Each of those coefficients to create individual functional maps. Even if you were to even if you were to test within a tiny integer range of let's say -10 to 10 >> and try all of the different coefficients, that would be something like 10 the 475 candidates. >> Right. >> Right. >> Right. even like a million cubit quantum computer if we were to create a quantum algorithm that could somehow do this which that's I'm just I'm just saying we don't even have one >> for another day.

1:31:00>> Yeah, that's for another day. We don't even have one. But like you you can't you can't do that. This is not a brute force thing, right? Like Fable had to understand something about the structure of polomial maps. understand something about um the properties of polomial maps