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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

#231

Post by jychang »

nserrino wrote: post #230
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…

For a^3 + b^3 + c^3 = 42, You can enumerate a,b pairs and then you need to check whether the "locked in" value of c^3 is a cube. However imagine it takes 1ns to validate a given pair [a,b]. The eventual solution was [-80538738812075974, 80435758145817515, 12602123297335631]. Since no combination of 3 positive (and therefore small) numbers has worked, we know that one of a,b,c are negative. Let's assume at least one o…

> I am interested to learn more about how they solved it if not brute force.

https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf

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

#233

Post by vkou »

dbancajas wrote: post #215
robinhouston wrote: post #102

I’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-ques...

Serious 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.

There 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.

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

#235

Post by apaprocki »

FreeFull wrote: post #194
kelnos wrote: post #193

Earlier quoted context omitted.

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)

I’m actually surprised someone at Google hasn’t switched to BigInt yet in JS if the browser supports it? Googlers, front page OKR time! :)

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

#236

Post by Wowfunhappy »

benologist wrote: post #204
dang wrote: post #196

Earlier quoted context omitted.

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.

The best solution would of course be instantaneously with some javascript but if it's not CPU-taxing it would still be plenty fast to send the title to the server as the input value changes. It's not uncommon for forms to do this with username eligibility checks etc.

Hacker News, unlike most websites, likely has a very significant number of users who keep Javascript disabled.

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

#237

Post by chrisau »

nserrino wrote: post #230
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…

For a^3 + b^3 + c^3 = 42, You can enumerate a,b pairs and then you need to check whether the "locked in" value of c^3 is a cube. However imagine it takes 1ns to validate a given pair [a,b]. The eventual solution was [-80538738812075974, 80435758145817515, 12602123297335631]. Since no combination of 3 positive (and therefore small) numbers has worked, we know that one of a,b,c are negative. Let's assume at least one o…

> I am interested to learn more about how they solved it if not brute force.

"Professors Booker and Sutherland's solution for 42 would be found by using Charity Engine; a 'worldwide computer' that harnesses idle, unused computing power from over 500,000 home PCs to create a crowd-sourced, super-green platform made entirely from otherwise wasted capacity."

https://phys.org/news/2019-09-sum-cubes-solvedusing-real-lif...

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

#238

Post by perl4ever »

cyphar wrote: post #214
seph-reed wrote: post #140

Earlier quoted context omitted.

While I do agree with you, I wouldn't be so quick to cast out numerology as a thing of importance. Hippies are pretty decent at sniffing things out, even if they explain them in totally insane ways. EDIT: If you're opposed to this comment, I request that you explain your reasoning. Making the assumption that happiness is of any value, there's actually quite a bit to dig into here.

Numerology is factually not a branch of mathematics. As for "hippies are pretty decent at sniffing things out", just because someone was correct about something in the past (despite a lack of evidence) does not make them correct today if they still lack evidence. The burden of proof is on numerologists to show that they are correct, not on everyone else to disprove it.

What is the reason for bringing up numerology? It was apparently prompted by the Wikipedia links, but I don't see why.

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

#239

Post by chrisau »

Ignoring 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.)

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

#240

Post by cyphar »

perl4ever wrote: post #238
cyphar wrote: post #214

Earlier quoted context omitted.

Numerology is factually not a branch of mathematics. As for "hippies are pretty decent at sniffing things out", just because someone was correct about something in the past (despite a lack of evidence) does not make them correct today if they still lack evidence. The burden of proof is on numerologists to show that they are correct, not on everyone else to disprove it.

What is the reason for bringing up numerology? It was apparently prompted by the Wikipedia links, but I don't see why.

The comment further up the chain (with the wikipedia links) was referencing the "importance" of 33 and 42 because they show up in quite a few fields -- assigning importance to certain numerical values is a kind of watered-down numerology (the fact that 42 was interesting in this problem doesn't necessarily mean that its "interesting-ness" transfers to the number itself).

(But I didn't bring it up, someone else did -- I was just responding to the argument that "there might be something to it".)

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