33 ms·
42 is found to be the sum of three cubes
- yumraj 7y agoHow was this computed? Just brute force ?
- techolic 7y agoSee also https://twitter.com/robinhouston/status/1169877007045296128 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.
- tspiteri 7y agoHow about "Forty-two"?
- benologist 7y agoJust page @dang because this is a filtering thing - https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html If the title begins with a number or number + gratuitous adjective, we'd appreciate it if you'd crop it. E.g. translate "10 Ways To Do X" to "How To Do X," and "14 Amazing Ys" to "Ys." Exception: when the number is meaningful, e.g. "The 5 Platonic Solids."
- yodon 7y agoPlease don't use code formatting for quotes. It makes the text unreadable on mobile.
- jacquesm 7y agoPlease don't criticize others for using features of the website you are visiting. If you bring a keyhole to view a landscape you won't get the full experience either. Rendering pages is a matter that should be solved by the client.
- snazz 7y agoIt seems to be readable on iOS Safari. Scrolling sideways works fine. Worse than a regular quote, but not actually that bad. It is a client issue if it’s more broken than that.
- rococode 7y agoSame on Android Firefox, although it is a bit annoying to have to scroll back and forth to read multi-line quotes formatted like this.
- tonmoy 7y agoText after a blank line that is indented by two or more spaces is reproduced verbatim. (This is intended for code.)
- deleted 7y ago[deleted]
- saagarjha 7y agoI don't see what the issue is with reminding people to be considerate of those who are browsing on mobile. Unfortunately, the latter group is also usually the ones with the least ability to actual solve their rendering problems.
- Dylan16807 7y ago
- techolic 7y agoInteresting, I didn't even think of changing to "Forty-two".
- yjftsjthsd-h 7y agoYou thought of using an obscure Unicode character before you thought of spelling it out? Well the good news is that you really do belong here on Hacker News :-)
- techolic 7y agoI seemed to be focusing on replacing 42 with a visual equivalence, I even started with ４２ then felt it looked quite ugly.
- bityard 7y agoOr "The Number 42..."
- hprotagonist 7y agoI spent a few seconds wondering if the 'circle-around-a-' was a ring operator of which I was previously unaware ...
- SideburnsOfDoom 7y agoI spent a while trying to work out if 42 in-a-circle was some kind of operator like 42! (factorial) or the like, because those three large numbers, cubed, didn't seem to refer to anything two-digit.
- sp332 7y agoThe title is sanitized on first submission, but you can edit the title afterward and put it back in. This works with some punctuation that gets cleaned out too.
- dang 7y agoOk, we've switched the URL from https://math.mit.edu/~drew/ https://math.mit.edu/~drew/ and removed the housing from 42 above.
- deleted 7y ago[deleted]
- saagarjha 7y agoHave you considered surfacing how titles will be changed when they're posted?
- dang 7y agoI'm not sure I understand your question, but we've posted countless times about this and are happy to answer about specific cases. You can find a lot of information via https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=by%3Adang%20title&sort=byDate&type=comment https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que... and similar searches.
- benologist 7y agoDuring the submission process where the form say a title is too long you could also be displaying how any transformations will affect it.
- saagarjha 7y agoYup, basically this. Another idea would be to add a "please review this title that Hacker News won't let me submit with" button that would get these kinds of things in front of you sooner.
- dang 7y agoThat shows up as soon as the submission is created, and there's an 'edit' link if you need to change it. Just don't change it in ways that break the site guidelines: https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html.
- gerdesj 7y agoI've just installed ttf-code2000 just so I could see that character in your comment!
- artie_effim 7y agochecks out https://www.wolframalpha.com/input/?i=%28-80538738812075974%29%5E3+%2B+80435758145817515%5E3+%2B+12602123297335631%5E3+ https://www.wolframalpha.com/input/?i=%28-80538738812075974%...
- thekyle 7y agoCould someone elaborate on why this is interesting? The linked page doesn't have much context.
- JshWright 7y agoNumberphile is a great resource for pretty much any "pop math" topic. https://www.youtube.com/watch?v=ASoz_NuIvP0 https://www.youtube.com/watch?v=ASoz_NuIvP0
- ArneVogel 7y agoThey also just released a video on 42: https://www.youtube.com/watch?v=zyG8Vlw5aAw https://www.youtube.com/watch?v=zyG8Vlw5aAw
- ajross 7y agoThe form of the statement sounds like something related to Fermat's last theorem, I guess?
- fl0wenol 7y agoIt is an open question as to whether every integer not equal to 4 or 5 modulo 9 is the sum of three cubes, although it is suspected to be true. https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computational_results https://en.wikipedia.org/wiki/Sums_of_three_cubes#Computatio... 33 and 42 were known to be exceptions of all sums less than 100 for which solutions were found, until recently. 42 was the most recent to fall.
- huntermonk 7y agoThanks, this is helpful. Why do we care if every integer not equal to 4 or 5 modulo 9 is the sum of three cubes? Just for fun?
- ipince 7y agoWell, yeah, pretty much. In number theory you study properties/patterns of the numbers. This is an example of such a property. These properties often seem like they don’t have any applications, and often they don’t, but not always. For example, cryptography is basically all based on number theory.
- rank0 7y agoI must be missing something. Why is this significant?
- qrybam 7y agoI suppose it’s in reference to https://en.m.wikipedia.org/wiki/Phrases_from_The_Hitchhiker's_Guide_to_the_Galaxy https://en.m.wikipedia.org/wiki/Phrases_from_The_Hitchhiker'...
- JshWright 7y agoIt was the last (possible) number under 100 to be confirmed as the sum of three squares.
- tobr 7y agoI believe it’s the lowest number that did not have a known solution for x^3 + y^3 + z^3. Plus, it’s 42.
- timerol 7y agoLowest number that isn't 4 or 5 modulo 9. There are numbers that can't be expressed, the open question is whether all other numbers can be expressed
- inscionent 7y agoSo long and thanks for all the fish.
- federicoponzi 7y agodocument.body.innerHTML = document.body.innerHTML.replace("<!--", "");
- deleted 7y ago[deleted]
- hombre_fatal 7y agoWhat exactly do you think "this" is? /~drew is their personal page and they probably hid it all to put this little gimmick as their homepage. It's commented out for easy reversal once it sails its course.
- airstrike 7y agoThanks, but why tho
- deleted 7y ago[deleted]
- deleted 7y ago[deleted]
- isotropy 7y agoFor context, 42 was the only remaining number below 100 where it wasn’t known if this was possible. The general problem of exactly which numbers are the sum of three cubes is unsolved. https://en.m.wikipedia.org/wiki/Sums_of_three_cubes https://en.m.wikipedia.org/wiki/Sums_of_three_cubes
- jjtheblunt 7y agoIsn't the set of all triples of integers be enumerable, because (inductively) the set of all pairs of integers is enumerable (thanks, Cantor)? Then, if one could enumerate all triples of integers, one could, for each triple, calculate the sum of the cubes. So, integers which are sums of cubes are enumerable. That doesn't mean they're a known recursive set, just recursively enumerable. Am I missing something? Perhaps the statement "The general problem of exactly which numbers are the sum of three cubes is unsolved." is meant to imply "not known to be a recursive set", i.e., no known algorithm for finite time answering the question of whether a given integer is in the set?
- travisoneill1 7y agoAll integer sets of any length would be enumerable, but that doesn't necessarily mean that one of them can be cubed and added/subtracted to make a given integer.
- jjtheblunt 7y agoAgreed, though enumeration of triples means you can enumerate sums of cubes of the triples...it just would take literally forever to enumerate them.
- baddox 7y agoYou can enumerate all syntactically valid proofs too, and just check each one to see if it proves whatever proposition you’re curious about.
- petrikapu 7y agoIn Python >>> (-80538738812075974)**3 + 80435758145817515**3 + 12602123297335631**3 42
- antoineMoPa 7y agoIn bash $ echo "print((-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3)" | sed "s/\^/**/g" | python - 42
- hazeii 7y agoIn calc: $ calc calc 2.12.4.1 > (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 42 > Literally pasted off the web page and verified in a couple of seconds.
- wruza 7y agoOne more proof that you can do everything in bash.
- saagarjha 7y agoWithout shelling out: $ echo $(((-80538738812075974)**3 + 80435758145817515**3 + 12602123297335631**3)) 42
- Franciscouzo 7y agoYou're just using python, using bc is more bashy: $ echo '(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3' | bc 42
- zests 7y agoIn dc/shell/bash dc -e '_80538738812075974 3 ^ 80435758145817515 3 ^ 12602123297335631 3 ^ + + p'
- taneliv 7y agoIn SQL: SELECT (-80538738812075974::NUMERIC)^3 + 80435758145817515::NUMERIC^3 + 12602123297335631::NUMERIC^3;
- mombul 7y agoYeah 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?
- deleted 7y ago[deleted]
- nkrisc 7y agoMost 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.
- scrungus 7y agoIf 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.
- sp332 7y agoGuessing you meant "trivial". Do you have some info about how this helps break encryption?
- scrungus 7y agoi would show you the proof but it's arbitrary because it only works against the "sum of cubes" encryption system, which only i use.
- saalweachter 7y agoThe math is complicated, but the TL;DR is that you should already be selling your Bitcoins. Dogecoin should last another six months, seven at the outside.
- deleted 7y ago[deleted]
- yati 7y ago
- rdiddly 7y agoIt's also the answer to life, the universe and everything. Which is nice.
- 98codes 7y agoThe ASCII code page index for it anyway.
- rawburt 7y agoIt's the answer to the ultimate "question" of life, the universe, and everything. This is an important difference. What's the question?
- paulmd 7y ago"What do you get when you multiply six by nine?"
- jaclaz 7y ago"I may be a sorry case, but I don't write jokes in base 13."
- paulmd 7y agoThe people who were looking for a trick are missing the point. The joke is that the answer doesn't work. What do you get when you multiply six by nine? Forty-two. That's the meaning of life. "We apologize for the inconvenience."
- arthurcolle 7y agoWas the author so excited by this finding that he replaced his academic page with just this expression? Very interesting finding in any case!
- deleted 7y ago[deleted]
- whitehouse3 7y agoWhy doesn't this evaluate in Excel? =(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 Returns 1.09785E+36
- outside1234 7y agoInteger overflow?
- dagw 7y agoExcel can't to math on arbitrarly large ints, so tuncates all the numbers and gives you the wrong result
- pfd1986 7y agohttps://support.office.com/en-us/article/set-rounding-precision-e5d707e3-07a8-4df2-810c-218c531eb06a https://support.office.com/en-us/article/set-rounding-precis...
- Tuna-Fish 7y agoBecause the numbers cubed are way larger than the numbers that Excel can represent precisely. Excel uses 64-bit IEEE 754 floats for all arithmetic. They can be used to accurately describe integers smaller than 2^53, or ~10^16. The first number cubed is ~-5*10^50, so you are very clearly outside the range where excel is precise. You should not use Excel to do any serious mathematics, or really anything else where numerical precision is important.
- colejohnson66 7y agoWhy is it that a program that people use for accounting doesn’t use arbitrary precision arithmetic?
- jz391 7y agoBecause for accounting the numbers easily fit in the supported range; even the US National Debt (https://www.usdebtclock.org/ https://www.usdebtclock.org/)...
- jliptzin 7y agoAny writeup for how this was found? Any interesting algorithm or was it just brute forced?
- 0-_-0 7y agoBrute forced with a smart formula. Here's a video for 33 (the previous 42): https://www.youtube.com/watch?v=ASoz_NuIvP0 https://www.youtube.com/watch?v=ASoz_NuIvP0
- binaural 7y ago42 is also the answer to the universe. ;-)
- intuitionist 7y agoThis is a great announcement; it reminds me of the legendary story of Frank Nelson Cole wordlessly announcing his factorization of 2^67 - 1: https://en.m.wikipedia.org/wiki/Frank_Nelson_Cole https://en.m.wikipedia.org/wiki/Frank_Nelson_Cole
- mzs 7y agoview the source, wasn't really wordless
- alanbernstein 7y agoIs that an HTML joke? The source says "At a mathematical meeting in New York in 1903, F. N. Cole walked on to the platform and, without saying a single word, wrote two large numbers on the blackboard. He multiplied them out in longhand, and equated the result to 2^67-1."
- pavel_lishin 7y agoWhich source?
- mzs 7y agoSorry, looks like the link has been change to a tweet from http://math.mit.edu/~drew/ http://math.mit.edu/~drew/ I meant the html source of Andrew's dept homepage titled "Life, the Universe, and Everything" at that moment.
- alanbernstein 7y agoWell, pretty close anyway, technically it's only one comment...
- mzs 7y agoand a new title $ curl -s https://math.mit.edu/~drew/ | python3 -c 'from sys import stdin as s; from html.parser import HTMLParser as P; p = P(); p.handle_data = print; p.feed(s.read())' | tr -s ' \n' Life, the Universe, and Everything (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 $
- jacquesm 7y agoThat's got to be the shortest article on HN ever to make the homepage.
- countryqt30 7y agoIf I copy and paste this formula to Google, I get a different result
- idbehold 7y agohttps://www.wolframalpha.com/input/?i=%28-80538738812075974%29%5E3+%2B+80435758145817515%5E3+%2B+12602123297335631%5E3 https://www.wolframalpha.com/input/?i=%28-80538738812075974%...
- saagarjha 7y agoGoogle seems to be suffering from a loss in precision: https://www.google.com/search?hl=en&q=(%2D80538738812075974)%5E3%20%2B%2080435758145817515%5E3%20%2B%2012602123297335631%5E3 https://www.google.com/search?hl=en&q=(%2D80538738812075974)...
- densetsu 7y agoHitchhikers rejoice!
- julienchastang 7y agoI think a lot of people are missing the Doug Adam’s reference. Was the The Hitchhiker's Guide to the Galaxy not the reading staple of young HN participants?
- mjklin 7y agoNow for some Vogon poetry
- JustSomeNobody 7y agoSo, does this mean life, the universe, and everything are all cubes?
- asdfman123 7y agoNo, it's just that Deep Thought misunderstood the question. It thought it was solving for the sum of three cubes.
- e12e 7y agoIt's a voxel world, I guess.
- robinhouston 7y agoI’ve just seen that my tweet is here! If anyone wants more details, I wrote a more detailed account, which has just been published at https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/ https://aperiodical.com/2019/09/42-is-the-answer-to-the-ques...
- dbancajas 7y agoSerious question, what is one of the immediate practical applications if this problem (sums of three cubes) is solved? Not a math person so question might sound stupid.
- hbosch 7y agoI think this falls under the mathematical category of "fun things you can do with numbers", and not much else. Just cool.
- Florin_Andrei 7y agoUnless in the future someone finds a useful application for this. It's happened before.
- pvinis 7y agoInteresting. Like what?
- crobertsbmw 7y agoPretty much all of mathematics was founded on "fun things to do with numbers", and then later we came up with using prime numbers for cryptography and finding Fibonacci sequences in nature, etc.
- vkou 7y agoThere aren't any, and the solution itself is uninteresting, but it's possible that techniques developed in the process of solving it will find application in solving other problems (In physics, chemistry, genetics, cryptography...) Consider the most important function of calculus - the integral. In layman's terms, it measures the area under a graph. Okay - that's a little bit useful, if you care about the physics of moving objects (A bus is accelerating at 2 m/s^2 for 5 seconds, how far does it travel..?) Yet, if you know how to integrate, a mountain of not-immediately obvious physics problems - say, anything that has to do with electromagnetism (Maxwell's equations) immediately become tractable.
- alejohausner 7y agoWhat about zero? Can zero be the sum of three cubes? Actually, I read that somebody proved it can't be done, but the proof is too long to fit in this comment. ;-)
- deleted 7y ago[deleted]
- 3JPLW 7y agoYes. 0³+0³+0³. Or x³+(-x)³+0³.
- FreeFull 7y agoFor this particular problem, none of the cubes are allowed to be 0
- OscarCunningham 7y agoI don't think that's a restriction that's usually enforced in this case. It seems nicer to allow the cubes to be zero since then the conjecture that the achievable sums are everything not 4 or 5 mod 9 can be stated without special cases.
- 3JPLW 7y agoYes. The joke the original commenter is playing is that Fermat's `x³ + y³ = z³` is equivalent to `x³ + y³ + (-z)³ = 0`.
- raghavkhanna 7y agoHere's a great video by numberphile on the subject https://www.youtube.com/watch?v=ASoz_NuIvP0 https://www.youtube.com/watch?v=ASoz_NuIvP0
- alskdj21 7y agoHere's a Numberphile video about this news. https://www.youtube.com/watch?v=zyG8Vlw5aAw https://www.youtube.com/watch?v=zyG8Vlw5aAw
- Scoobert 7y agoClicked hoping for Douglas Adams, got Numberwang
- julienchastang 7y agoExactly! In case everyone has forgotten 42 is the "Answer to the Ultimate Question of Life, the Universe, and Everything" [1] [1] https://en.wikipedia.org/wiki/42_(number)#The_Hitchhiker's_Guide_to_the_Galaxy https://en.wikipedia.org/wiki/42_(number)#The_Hitchhiker's_G...
- psv1 7y agoThis is one of those things that has been mentioned, used and worn out so much that you feel much cooler when you don't acknowledge it.
- oo0shiny 7y agoYou come to HN to be cool!?
- mywittyname 7y agoWhere do you go to be cool?
- MFLoon 7y agoNo, it's just that not mentioning it is has a cool factor of 0, neither cool nor uncool, and mentioning it has a cool factor of around negative infinity.
- young_mete 7y agoHappy someone mentioned it. I almost got desperate scrolling through the comments...
- mdtancsa 7y agoFunny, I was hoping for NumberWang, or at least the German version...
- sabujp 7y agoi love bc, just to verify : bc 1.07.1 Copyright 1991-1994, 1997, 1998, 2000, 2004, 2006, 2008, 2012-2017 Free Software Foundation, Inc. This is free software with ABSOLUTELY NO WARRANTY. For details type `warranty'. (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 42
- airocker 7y agoCan this be used instead of RSA?
- gmiller123456 7y agoNo. The key to RSA is that it is relatively easy to find two primes and multiply them together. With the sum of three primes, there is no easy problem to be solved which will be harder for someone else.
- deleted 7y ago[deleted]
- djyaz1200 7y agohttp://youtu.be/8190ziL5v-k http://youtu.be/8190ziL5v-k
- AstralStorm 7y agoAnd the universe turned slightly weirder again.
- sidcool 7y agoAny idea what algorithm was used? Was it brute forcing through all combinations?
- gunnihinn 7y agoAs a mathematician and long-time sufferer of that Douglas Adams meme: GIVE A FUCK?
- archie2 7y agoWhy should I care?
- t_minus_3 7y agodumb question: What is the practical application of this conjecture?
- ivan_ah 7y agoWe can easily verify using SymPy on live.sympy.org: https://live.sympy.org/?evaluate=(-80538738812075974)**3%20%2B%2080435758145817515**3%20%2B%2012602123297335631**3%0A%23--%0A https://live.sympy.org/?evaluate=(-80538738812075974)**3%20%...
- mzs 7y ago$ lynx -dump math.mit.edu/~drew | bc 42 $
- Lio 7y ago_randomly pulling letters out of a scrabble bag_ (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 is 42? 42?! I always said there was something fundamentally wrong with the universe.
- dustinmr 7y agoNow that we have the answer, I wonder what the question was? And why is that mouse looking at me?
- chrisau 7y agoIgnoring the century in the year, I love the coincidence that this question was first asked in the year '54 and in "Hitchhiker's", it's said that 6x9 and 42 are the same. In fact they are if 6x9 is base 10 (6x9=54), and 42 is base 13 (4x13+2 = 54). So, we get three coincidences: This question was asked in '54 ; "Hitchhiker's" said the answer to the meaning of life is 54 (6x9, 42 base 13); and best of all, 42 was the last number solved. And if you want to add one more coincidence, this was solved in the year '19 (connecting 19 and 54). All irrelevant coincidences applied with hindsight, but fun. (Douglas Adams claimed that 42 was a randomly chosen number, but I'd argue his subconscious had been processing the idea for a while and gave him a number with a meaning. We just don't know which is the correct meaning.)
- brokenmachine 7y agoI understand that it is interesting, but can someone ELI5 what this might be useful for in real life?
- CharityEngine 7y agoHey guys, Mark from Charity Engine here. If anyone else could use planetary-scale computing (potentially for free, if your project is really cool), then come and talk to us. The CE grid has over 2 million CPU cores, 600k+ unique IP addresses, can run anything in a Docker container and has integration with Ethereum. Adding Kubernetes and GPU support as we speak. You can fire it up with 8 clicks, easier than AWS. This is literally a fully-functional worldwide computer. PS. It's not just ridiculously huge, it is also ridiculously cheap :)
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