# 42 was the last remaining number below 100 which could not be expressed as the sum of three cubes - until now

Originally published at: https://boingboing.net/2019/09/06/736626.html

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I particularly liked that in order to find the correct “question” to solve for 42 that they had to replace the supercomputer they used for 33 with a computer network that was, in a certain sense, the size of the Earth. Sounds familiar.

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I’m assuming this means that the answer to the ultimate question of life, the universe and everything is no longer 42, and the whole simulation was a failure. Thanks Slartibartfast.

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It’s interesting (and the tie-in to Hitchhiker’s Guide always welcome since I’ve used 42 in my avatar names since the dawn of the WWW), but can someone tell me why it matters? What does it do for us that we can now describe all numbers below 100 as the “sum of three cubes?” Just an interesting mathematical theorem? The development of a new sub-atomic physics? A way to reduce gaseous emissions from mammalian species?

EDIT: I’ll add it’s totally cool if it does NOT matter other than “it is, and that’s lovely.” Just curious.

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Beschizza knew this an hour before everyone else. He’s ahead of his time… ever so slightly.

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There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.

There is another theory mentioned, which states that this has already happened

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DON’T PANIC

(Yes, stupid rules, this was all I wanted to say.)

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that life has no meaning anymore. this timeline does not compute. divide by zero.

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They’re currently building a second computer to determine why humans were compelled to find all the numbers that could be expressed as the sum of three cubes.

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Three cubes:

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After watching that video, I still have no idea why this is significant. If it had a cryptographic application, I could at least understand the hoopla.

42 was Jackie Robinson’s number with the Dodgers, and has been retired from use across all of Major League Baseball.

42, in and of itself, is greater a number than the sum of its cubes…

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How do they know that for some of the numbers that there is no solution? Is there a proof that us mere mortals can comprehend?

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Me: “Wait, that’s a trivial problem… isn’t it? Smart people have been working on this since 1954, I can’t be seeing the real problem. Oh! Cubes of negative numbers! Aaargh!”

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I’m going to guess that someone figured this out say back in November 2016…

(it’s the only way I’ve got to explain the current timeline)

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It really doesn’t “matter” in the sense of having an application, no. It is just interesting in that what seems to have been a low-hanging fruit has been unsolved until now.

Cube a handful of integers and you’ll notice that they are either divisible by 9 or off by 1 from a number that is.
You can verify this with some algebra by expanding (3k+1)^3 and finding a common factor of 9 in all the terms except the dangling +1 (similarly in the -1 case, and the (3k)^3 case is trivial enough).

Summing three cubes will get you at most 3 away from a multiple of 9. The numbers deemed “impossible” in the video are all 4 away. Example, 31 is 27 + 4; 32 is 36 - 4.

Hope that helps!

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I thought purplecat replied to this post which I think is just as appropriate:

Here’s Numberphile on the significance of this discovery.

Did you link to the wrong video Mark? They didn’t explain the significance, like at all.

Why is touched on obliquely in the numberphile video. It’s a hard problem. Writing fast algorithms to pick all sets of three numbers from an arbitrarily large set and developing ways of eliminating ones that you know aren’t going to work is mathematically interesting in itself. More useful is finding a way to get the computer power you need. Firstly getting enough so that it takes a human scale amount of time to do the search. And then getting whoever owns it to let you use it for as long as you need. Of these three problems, the last one seems the hardest. Although they seem to have solved it and the second one is the hardest again. They may get lucky with 114, but they’re probably just going to have to wait.

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