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42 is found to be the sum of three cubes
141–150 of 256 posts
Re: 42 is found to be the sum of three cubes
#142In Python >>> (-80538738812075974)**3 + 80435758145817515**3 + 12602123297335631**3 42
In Google:
1.9892987e+35
Re: 42 is found to be the sum of three cubes
#143Earlier quoted context omitted.
view the source, wasn't really wordless
Which source?
Sorry, looks like the link has been change to a tweet from http://math.mit.edu/~drew/ I meant the html source of Andrew's dept homepage titled "Life, the Universe, and Everything" at that moment.
Re: 42 is found to be the sum of three cubes
#144Why doesn't this evaluate in Excel? =(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 Returns 1.09785E+36
Welcome to numerical computing :D! Btw: julia> (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 42
Or
CL-USER> (+ (expt -80538738812075974 3) (expt 80435758145817515 3) (expt 12602123297335631 3))
42Re: 42 is found to be the sum of three cubes
#145Earlier quoted context omitted.
Isn'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…
All 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.
Agreed, though enumeration of triples means you can enumerate sums of cubes of the triples...it just would take literally forever to enumerate them.
Re: 42 is found to be the sum of three cubes
#146Earlier quoted context omitted.
Yes. 0³+0³+0³. Or x³+(-x)³+0³.
For this particular problem, none of the cubes are allowed to be 0
I 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.
Re: 42 is found to be the sum of three cubes
#147Re: 42 is found to be the sum of three cubes
#148See 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.
Ok, we've switched the URL from https://math.mit.edu/~drew/ and removed the housing from 42 above.
why did you switch from the mit site to twitter ? (i'm currently blocked on twitter - they want a phone number and are blocking all content till i provide one)
Re: 42 is found to be the sum of three cubes
#149Earlier quoted context omitted.
For this particular problem, none of the cubes are allowed to be 0
I 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.
Yes. The joke the original commenter is playing is that Fermat's `x³ + y³ = z³` is equivalent to `x³ + y³ + (-z)³ = 0`.
Re: 42 is found to be the sum of three cubes
#150Could someone elaborate on why this is interesting? The linked page doesn't have much context.
It’s a problem that is easy to state, yet hard to fully solve, and doesn’t seem like it should be unsurmountable, as it is easy to make some progress on (the “integers that equal 4 or 5 modulo 9 can’t be written as the sum of 3 powers of integers” part handles 22% of cases, and interested kids can understand it)
That makes it interesting to most mathematicians (if it weren’t easy to state, it could still be interesting, but to a smaller audience)
Edit: it also is easy for non-mathematicians with a knack for computing to get a crack at. You need not know much about number theory to write a program that searches for solutions for the unknown cases (it will help, but knowing how to optimize programs may help more, and that’s not something all mathematicians are good in). That grows the set of people who might find this interesting.