Today I'd like to revisit some really big numbers we've covered recently on Numberphile
and then maybe reflect on how big they really are.
Now, it all starts with this video we made about the sum of three cubes.
[Past video] And this is a sort of area of
research that I'm, I'm actively engaged with. - Tim Browning told us how all numbers could be expressed as the sum of three cubes.
For example, ten can be represented as two cubed, plus one cubed,
plus one cubed. There you go, the sum of three cubed numbers.
Eleven can be three
minus two
and minus two. That's right, you can use negative numbers as well.
Seventeen is as simple as 2,
2, and 1; while 45 can be 4, minus 3, and 2.
Now look I've already told a little fib here, because not all integers can be expressed as the sum of three cubes.
In fact, it's been proven that numbers of these forms can't be, meaning these integers will never work.
You'll never get them as the sum of three cubes. Sorry about that. But all these other ones should have a solution.
I *say* should but here's a problem, at the time we made the video
there were three numbers below 100 that should have been possible, that hadn't been found.
33, 42 and 74.
A Numberphile viewer, Sander Huisman, let a powerful computer loose on the problem and he cracked 74.
Look at this - three absolute telephone numbers.
But that still left 33 and 42.
Andrew Booker from the University of Bristol refined the algorithms and used this supercomputer to solve 33.
Here's what he found,
some real whoppers again. That left 42, but with some extra help - you can guess what happened?
Here it is.
42 cracked as well. Still no rigorous proof
that all numbers are possible, that glorious moment still awaits mathematics.
But at least those three glaring elephants in the room below 100 were removed. But here's something else I didn't tell you,
not only do mathematicians think all integers are possible; they think they're possible in an infinite number of ways.
Take 10 for example. I've already showed you 2, 1 and 1,
but there's also 4, -3, and -3.
And these ones as well, and as I said, in all likelihood the options continue forever.
So this solution for 42, that just happens to be the first one among the number line.
Let the computers search for long enough and they'll find another, and
another, and another. And that's going to bring us to number 3 - because you see,
the number 3 is a bit of a pin-up in this area of research. Since mathematicians first started dealing with this problem
it's been used to exemplify the whole conundrum.
Now I'm sure you can think of this one very obvious solution,
1, 1 and 1. And the next solution isn't too tricky either - minus five,
four, and four.
But if this whole infinite solution thing is true, where's the next one?
It was so hard to find, this famous paper on the topic noted "it was very difficult indeed", classic understatement.
So Andrew here decided the third solution for 3, that was his new holy grail.
Again, they unleashed the computers and you can guess what's coming - here it is everyone:
The next solution for 3. Now, let's just focus on that first number, the positive one. It's got 21 digits.
Just how big is that cube? I think we can easily take a number like this for granted.
Now you've been seeing me make some of these numbers as cubes here with wooden blocks. If I had more of them,
like a lot more of them,
what if I was to make that big number that I just showed you for the number 3?
I'm gonna need a lot more of them.
Would this new cube stretch into space? Well, yes. Would it reach the moon or even the sun?
Certainly.
Would it reach the nearest star? That's some 4.2 light years away
It would.
In fact, here's a visualisation of the milky way.
Our massive sun, our big star at the center of the solar system,
well, in this context it's just a pretty boring old speck of a star sitting here amongst over a hundred billion others.
You can't even really distinguish it.
But if we built that epic cube, and started it here in my office,
this is what it would look like.
It'd dominate the galaxy. It'd be clearly visible from a neighbour like Andromeda. By the way,
it would also weigh more than all the other atoms in the observable universe,
but that's another story. And that number, it forms just part of the third solution
for the number 3. And remember in all likelihood there are an infinite number of solutions
that are infinitely bigger!
Think how big these numbers must get. We often marvel at the size,
the epicness of space, of the universe.
But look at the epicness here in numbers and mathematics.
This was just a basic math problem, something as simple as the sum of three cubes and it's blown our minds.
If you'd like to see more mega favourite numbers from people here on Youtube,
I'm going to put a link on the screen and down in the description to a whole bunch of things
we've been doing, and maybe you can get involved too.
And i'll also include links to all the videos we've made about the sum of three cubes.