when a number can be written as a sum of three cubes

[REWIND] We still don't know the answer to that one so we've not yet been able to find any integers which when

we summed their cubes you get 33.

Since then we've had some breaking news! There's a paper that appeared and someone has actually

gone away and this guy's Sander Huisman, he's gone away and he calculated a new solution to one

integer that was previously unknown whether it was a sum of three cubes or not.

Well indeed he credits the Numberphile video, he sort of found that it was kind of interesting enough to just

have a go at pushing the boundaries of knowledge a little bit further and he managed to sort of get this going on.

A computer search, and I think he was searching for something like 12.5 CPU years on a bank of computers in

Lyon and he was able to find a new solution for the number 74.

So we are interested in, suppose that we're given an integer k. Can you find integers x, y and z, and these can

be both positive and negative such that x³ + y³ + z³ is equal to k?

If we just look at all of the integers between 1 and 99, as of last time there were only three integers on that list

for which we didn't have an answer about whether they can be written as a sum of three cubes or not.

And those were 33, 42 and 74.

Now we know that this last number here, 74 is actually represented as a sum of three cubes.

Do you want me to write it down for you?

Of course.

So here we go.

So 74 is now known to be equal to minus 284 650 292 555 885 cubed

plus 6 229 832 190 556 cubed

plus 283 450 105 697 727 cubed.

And there's the new solution, he also found some new solutions for larger values of k, but kind of the

interesting thing is that for for k less than a hundred we now only have two more numbers to worry about here.

The number 33 and the number 42.

And are they solvable? Are they out there? Should other people be firing up their computers?

Well absolutely, so far, thanks to Sander we now know that if we're interested in the number 33,

there are certainly no solutions which the size of x, y and z are bounded by 10 to the 15, so that's a pretty big

range, so you're gonna have to look further than that.

Who knows, it may will be that if you just look up 10 to the 16 you'll find a solution.

And 42? Do we know any more about that or

No, nothing about that

It's not earth-shattering news, I mean if we thought for a long time that these things do have solutions in fact we

think that there are infinitely many solutions for each of these values of k. Our expectation is that they are very

very sparse, so you might come across one and then the next time you come across another one

might be many many years down the line before you've got computers big enough to find them

So I mean it's gratifying that this kind of coinciding with our belief that these things do indeed have

solutions, but in terms of sort of fundamental mathematics, I would say that's not the case

it's kind of just a more extensive computer search.

Not every Numberphile viewer is going to make a contribution like Sander,

but the people you see listed on the screen have helped us out.

They're among our Patreon supporters and we really appreciate it.

If you'd like to join their ranks with no matter how small a contribution,

You can go to patreon.com/numberphile

I'll put a link in the video description.

And by the way, you can see on the screen at the moment

a link to our original video, which inspired Sander's investigation.

But we've also got a new video, and that's all about those values of k

for which it's impossible to find a solution. Not even Sander's gonna find one for that.

And if you'd like to see the proof,

We've just put it on Numberphile2.

And by the way, while you're over on Numberphile2,

Why not subscribe to the channel? Because I don't always mention it here,

and you might occasionally stumble over something you didn't even know existed.