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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

#181

Post by dang »

benologist wrote: post #154
dang wrote: post #152

Earlier quoted context omitted.

I'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&que... and similar searches.

During the submission process where the form say a title is too long you could also be displaying how any transformations will affect it.

That 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.

Re: 42 is found to be the sum of three cubes

#182

Post by colejohnson66 »

Tuna-Fish wrote: post #54
whitehouse3 wrote: post #39

Why doesn't this evaluate in Excel? =(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 Returns 1.09785E+36

Because 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 nume…

Why is it that a program that people use for accounting doesn’t use arbitrary precision arithmetic?

Re: 42 is found to be the sum of three cubes

#183

Post by mzs »

alanbernstein wrote: post #163
mzs wrote: post #143

Earlier quoted context omitted.

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.

Well, pretty close anyway, technically it's only one comment...

and 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
  $

Re: 42 is found to be the sum of three cubes

#184

Post by nwallin »

huntermonk wrote: post #90
fl0wenol wrote: post #19

Earlier quoted context omitted.

It 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#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.

Thanks, 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?

Probably.

It's possible that solving this problem in a satisfactory way would teach us something useful about number theory, that's useful in real world practical stuff.

Andrew Wiles' proof of Fermat's Last Theorem is based on the modularity of elliptic curves. I'm not sure to what extent Wiles' proof contributed to elliptic curve cryptography hitting the scene a decade later, but it probably didn't hurt.

Re: 42 is found to be the sum of three cubes

#185

Post by grzm »

dang wrote: post #181
benologist wrote: post #154

Earlier quoted context omitted.

During the submission process where the form say a title is too long you could also be displaying how any transformations will affect it.

That 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 .

I think they're getting at transformations that happen after submission, such as dropping initial question words and the like, not just the length restriction (which does appear).

Re: 42 is found to be the sum of three cubes

#186

Post by erikig »

ColanR wrote: post #178
jjtheblunt wrote: post #103

Earlier 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…

It is enumerable, but I think what you may have missed is that negative numbers are included. i.e., > 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

Strangely, typing this into Google does NOT result in 42

((-80538738812075974)^3)+(80435758145817515^3)+(12602123297335631^3)

Re: 42 is found to be the sum of three cubes

#187

Post by benologist »

dang wrote: post #181
benologist wrote: post #154

Earlier quoted context omitted.

During the submission process where the form say a title is too long you could also be displaying how any transformations will affect it.

That 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 .

If it displays during the title entry process it is educating the users about the transformations that occur too. It's a positive-reinforcement loop that supplements the guidelines. It's the reason you don't have to document in the guidelines that the maximum title length is 80 characters.

Re: 42 is found to be the sum of three cubes

#188

Post by defnmacro »

earenndil wrote: post #97
rawburt wrote: post #67

Earlier quoted context omitted.

No, it's the answer to the ultimate question of life, the universe, and everything.

No, it's the ultimate answer to the ultimate question of life, the universe, and everything.

So long and thanks for all the fish.

Re: 42 is found to be the sum of three cubes

#189

Post by mywittyname »

oo0shiny wrote: post #177
psv1 wrote: post #138

Earlier quoted context omitted.

This 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.

You come to HN to be cool!?

Where do you go to be cool?

Re: 42 is found to be the sum of three cubes

#190

Post by jz391 »

colejohnson66 wrote: post #182
Tuna-Fish wrote: post #54

Earlier quoted context omitted.

Because 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 nume…

Why is it that a program that people use for accounting doesn’t use arbitrary precision arithmetic?

Because for accounting the numbers easily fit in the supported range; even the US National Debt (https://www.usdebtclock.org/)...
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