I think that means 114 is now the smallest unsolved case.
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News! 42 is the sum of three cubes. http://math.mit.edu/~drew/ 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3
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This was found by Andrew Booker at Bristol and Andrew Sutherland at MIT.
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The remaining unsolved cases up to 1000, as far as I know, are: 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975.
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Every cube is within one of a multiple of nine, which means every sum of three cubes is within three of a multiple of nine. So numbers of the form 9k+4 or 9k+5 cannot be expressed as the sum of three cubes. It’s conjectured that every other whole number can be.
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I’ve written a bit more about this discovery at the request of
@aperiodical:https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/ …Diesen Thread anzeigen -
And
@numberphile have released an interview with Andrew Booker! https://youtu.be/zyG8Vlw5aAw He also mentions there that he has solved 795, so we can delete that from the list of unsolved cases.Diesen Thread anzeigen
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What sort of algorithms are used to search for these?
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We don’t know for sure about this one, but Andrew Booker (one of the two people who found this) previously used a fairly simple algebraic method, which he described in https://arxiv.org/abs/1903.04284 and also explained in a Numberphile videohttps://youtu.be/ASoz_NuIvP0
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Earlier searches were done using a ludicrously clever algorithm invented by Noam Elkies. https://arxiv.org/abs/math/0005139 …
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Exciting! Do you think you'd be able to write a couple of paragraphs about this news for an
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I don’t have any inside information. I’m just sharing the news that was posted on the Andrews’ web sites, which came to my attention via Math Overflow. But sure I can, if you don’t mind that.
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It'd be good to get a tiny amount of background/context, and you can probably do that easier than we can from a standing start! You could email something to root@aperiodical.com if easiest, or we can set you up as an author?
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Yep. Sure thing. I’ll send you an email. I’ll try and do it quickly while the news is hot, but I have some other things to do so it may be a few hours.
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You have new mail.
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Just post it already
@aperiodical@stecks we are waaaaitingpic.twitter.com/03LrFOkajW -
42 is the answer to the question “what is (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³" Thanks
@robinhouston@aperiodicalhttps://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/ …
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I dont think this is surprising, can’t you do this with every number not only with 42? For me this is the equivalent of taking a bigger number and negating it with a somewhat smaller number, so the difference is exactly 42
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You definitely can’t do it with *every* number. It can’t be done with 5, for example.
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Well, maybe you haven't tried enough numbers, lazybones
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Every cube of a number, mod 9, is either -1 or 1. So the sum of three cubes, mod 9, could be {-3, -2, -1, 0, 1, 2, 3} 5 mod 9 isn't in that range. Therefore, no sum of three cubes can equal 5.
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5 cubed is 125. 125mod(9) is 8. Am I missing something here? Also why does mod 9 matter?
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8 is -1, modulo 9. It matters because it’s the only such obstruction to the expressibility of an integer as the sum of three cubes.
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Thank you! I learned something today.
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