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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

#191

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

Ah, good old integer rounding...

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

#193

Post by kelnos »

erikig wrote: post #186
ColanR wrote: post #178

Earlier quoted context omitted.

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)

DDG doesn't handle it correctly either (gives the same very-large answer as Google). I assume there's some sort of integer overflow going on somewhere. Wolfram Alpha does ok, though: https://www.wolframalpha.com/input/?i=%28%28-805387388120759...

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

#194

Post by FreeFull »

kelnos wrote: post #193
erikig wrote: post #186

Earlier quoted context omitted.

Strangely, typing this into Google does NOT result in 42 ((-80538738812075974)^3)+(80435758145817515^3)+(12602123297335631^3)

DDG doesn't handle it correctly either (gives the same very-large answer as Google). I assume there's some sort of integer overflow going on somewhere. Wolfram Alpha does ok, though: https://www.wolframalpha.com/input/?i=%28%28-805387388120759...

Google and Duckduckgo use floating point rather than arbitrary precision integers. Wolfram Alpha's solver is significantly more advanced (including arbitrary precision for integers and floating point numbers)

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

#196

Post by dang »

benologist wrote: post #187
dang wrote: post #181

Earlier quoted context omitted.

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.

The title processing logic is surprisingly complex, just as the whole issue of titles on HN is surprisingly complex. That means the code runs on the server, which would make it awkward to run during the data entry process, which of course is browser-side.

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

#197

Post by MFLoon »

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!?

No, 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.

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

#198

Post by andrepd »

jjtheblunt wrote: post #103
isotropy wrote: post #21

For 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

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…

So what you're saying is (with needlessly complicated terms) that we can brute force the search for sums of three cubes?

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

#200

Post by jjtheblunt »

kmill wrote: post #166
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…

Many questions about the integers are recursively enumerable (like: what are the counterexamples to Fermat's last theorem?), but this doesn't really reveal anything about any deeper structure. Part of the conjecture is, indeed, that the set of numbers that are a sum of three cubes is recursive, which is given by a simple condition on the residue modulo 9 (see a sibling comment). However, mathematicians don't usually…

Agreed
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