How was this computed? Just brute force ?
42 is found to be the sum of three cubes
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Re: 42 is found to be the sum of three cubes
#32See also https://twitter.com/robinhouston/status/1169877007045296128 . I was prompted the twitter link has been submitted but wasn't able to find it. Edit: And apologies for using unicode ㊷ in the title, the ascii 42 was removed from the title after initial submission.
I spent a few seconds wondering if the 'circle-around-a-' was a ring operator of which I was previously unaware ...
Re: 42 is found to be the sum of three cubes
#33Yeah but now, what? What was learned from this? Why did we look for a solution in the first place? How is it important that a number can be a sum of three cubes?
Most mathematic problems, divorced from any other context, seem pointless.
But mathematics is the foundation for so many fields that it's worth coming to explore it.
Re: 42 is found to be the sum of three cubes
#34It's also the answer to life, the universe and everything. Which is nice.
Re: 42 is found to be the sum of three cubes
#35Yeah but now, what? What was learned from this? Why did we look for a solution in the first place? How is it important that a number can be a sum of three cubes?
If you can break up a reasonably-sized table (perhaps 120 or so) of numbers into their respective sums of cubes, breaking many types of encryption is arbitrary.
Re: 42 is found to be the sum of three cubes
#36Was the author so excited by this finding that he replaced his academic page with just this expression?
Very interesting finding in any case!
Re: 42 is found to be the sum of three cubes
#37Re: 42 is found to be the sum of three cubes
#38[deleted]
Re: 42 is found to be the sum of three cubes
#39Why doesn't this evaluate in Excel?
=(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3
Returns 1.09785E+36
Re: 42 is found to be the sum of three cubes
#40I must be missing something. Why is this significant?
Same question. People seem to be researching that kind of stuff (numbers that are sums of 3 cubes). Why?
I only recently heard what I think is a pretty solid answer to this question [1] that put it this way: We've pretty much got addition nailed down. We're pretty solid on multiplication. However, we are surprisingly weak mathematically when we try to put the two together; you can write down some extremely small math statements that baffled mathematicians for centuries, and for such statements that we have solutions for, they often get into very, very complicated math.
So you have things like Fermat's Last Theorum, or if you prefer, the Question: For a^n + b^n = c^n, are there any solutions with n > 2 for integers? Note how it's just multiplication and addition on the integers. There are simpler questions, but they don't get much simpler. There's no real analysis, transcendental numbers, imaginaries or higher-dimensional numbers, infinities, and so on, at least not in the question itself; it's literally so simple you can reasonably explain it to a middle schooler. It took mathematicians centuries, and the simplest version of the current proof is still PhD-level work to understand. (I suspect even a Master's student in math would have to carefully craft their entire post-doc education for the purpose of understanding this exact proof to be able to say they fully understood it, with no lemmas taken on faith, and I'm still not sure they could make it without a lot of independent study.)
You have things like "Can all of the numbers of a certain form be created via 'a^3 + b^3 + c^3'?", which sounds like it really ought to be simple. It's not too hard to eliminate 4 or 5 mod 9, but proving that it can always be done for everything else is beyond all current known math.
There's a lot of these sorts of questions in math right now. This is a particular single example of a family of very vexing problems.
It's hard to know what the practical impact of mastery over this combination would be, but I expect there probably would be some. Mastery over addition and multiplication have practical consequences that would require a series of books to explore; you have to think that mastering them in combination couldn't help but be practically useful.
[1]: I think it was a somewhat recent Numberphile video, but it may also have been a Terence Tao lecture. References solicited if you've got 'em.