
42 is found to be the sum of three cubes - techolic
https://twitter.com/robinhouston/status/1169877007045296128
======
isotropy
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](https://en.m.wikipedia.org/wiki/Sums_of_three_cubes)

~~~
jjtheblunt
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 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?

~~~
nserrino
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 of a,b are
negative since it doesn't matter how we allocate them.

To reach the final pair of a = 80435758145817515 (the smaller positive
integer) and b = -80538738812075974, you have to increment "a" (starting from
0) 80435758145817515 times and decrement "b" (starting from 0)
80538738812075974 times.

That is 80538738812075974*80435758145817515 possible combinations.

Let's assume each one takes 1 ns (which I believe is fairly optimistic at
least for a single machine)

That results in a runtime of 6.5e+24 seconds, aka 2.1e+17 years. No matter how
many machines you add, the brute force approach does not appear to be
feasible.

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

~~~
chrisau
> 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...](https://phys.org/news/2019-09-sum-cubes-solvedusing-real-life.html)

~~~
MertsA
>SUSTAINABLE, ULTRA-LOW CARBON - using PCs that already exist but are just
underused

Wow, do they honestly believe their own marketing nonsense? There's no way the
PUE and power efficiency of a bunch of old random desktop computers is going
to come close to beating a modern Amazon, Google, or Microsoft datacenter.
Cost wise, yeah sure, I'd bet it would be cheaper even with the lower
efficiency and increased power usage but as far as "super-green" and low
emissions this is just absurd. I think they might honestly not know that idle
power usage is a small fraction of full load usage for any modern processor.

~~~
CharityEngine
We don't need the datacenters at all. That's the difference.

No facilities. No hardware. No bricks, mortar, shipping, mining of metals or
rare earths. No replacing of millions of obsolete machines every three years.
We tread lighter than any datacenter owner can ever dream of

------
robinhouston
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...](https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-
what-
is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/)

~~~
dbancajas
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.

~~~
hbosch
I think this falls under the mathematical category of "fun things you can do
with numbers", and not much else. Just cool.

~~~
Florin_Andrei
Unless in the future someone finds a useful application for this. It's
happened before.

~~~
pvinis
Interesting. Like what?

~~~
crobertsbmw
Pretty 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.

------
techolic
See 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.

~~~
dang
Ok, we've switched the URL from
[https://math.mit.edu/~drew/](https://math.mit.edu/~drew/) and removed the
housing from 42 above.

~~~
nqzero
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)

~~~
Svip
Can't you just log out? Or are you IP blocked?

~~~
nqzero
given that they blocked me for "rules violations" (which i clearly didn't do,
as it was a brand new account that had taken zero actions and my previous
account was fine when i closed it) i didn't want to circumvent the block
(which might have actually been against the rules) so i didn't use technical
measures (which certainly would have worked). i ended up just giving them my
phone number, but it's a sad reality that we've accepted all these walled
gardens

------
intuitionist
This 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
view the source, wasn't really wordless

~~~
pavel_lishin
Which source?

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

~~~
mzs
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
      $

------
thekyle
Could someone elaborate on why this is interesting? The linked page doesn't
have much context.

~~~
JshWright
Numberphile 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
They also just released a video on 42:
[https://www.youtube.com/watch?v=zyG8Vlw5aAw](https://www.youtube.com/watch?v=zyG8Vlw5aAw)

------
federicoponzi
document.body.innerHTML = document.body.innerHTML.replace("<!--", "");

~~~
airstrike
Thanks, but why tho

------
ivan_ah
We can easily verify using SymPy on live.sympy.org:

[https://live.sympy.org/?evaluate=(-80538738812075974)**3%20%...](https://live.sympy.org/?evaluate=\(-80538738812075974\)**3%20%2B%2080435758145817515**3%20%2B%2012602123297335631**3%0A%23--%0A)

~~~
mzs

      $ lynx -dump math.mit.edu/~drew | bc
      42
      $

------
whitehouse3
Why doesn't this evaluate in Excel?

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

Returns 1.09785E+36

~~~
Tuna-Fish
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 numerical precision is important.

~~~
colejohnson66
Why is it that a program that people use for accounting doesn’t use arbitrary
precision arithmetic?

~~~
jz391
Because for accounting the numbers easily fit in the supported range; even the
US National Debt
([https://www.usdebtclock.org/](https://www.usdebtclock.org/))...

------
jacquesm
That's got to be the shortest article on HN ever to make the homepage.

------
alejohausner
What 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. ;-)

~~~
3JPLW
Yes. 0³+0³+0³. Or x³+(-x)³+0³.

~~~
FreeFull
For this particular problem, none of the cubes are allowed to be 0

~~~
OscarCunningham
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.

~~~
3JPLW
Yes. The joke the original commenter is playing is that Fermat's `x³ + y³ =
z³` is equivalent to `x³ + y³ + (-z)³ = 0`.

------
alskdj21
Here's a Numberphile video about this news.
[https://www.youtube.com/watch?v=zyG8Vlw5aAw](https://www.youtube.com/watch?v=zyG8Vlw5aAw)

------
Lio
_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.

------
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.)

------
CharityEngine
Hey 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 :)

------
artie_effim
checks out
[https://www.wolframalpha.com/input/?i=%28-80538738812075974%...](https://www.wolframalpha.com/input/?i=%28-80538738812075974%29%5E3+%2B+80435758145817515%5E3+%2B+12602123297335631%5E3+)

------
jliptzin
Any writeup for how this was found? Any interesting algorithm or was it just
brute forced?

~~~
0-_-0
Brute 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)

------
rank0
I must be missing something. Why is this significant?

~~~
ocfnash
Mathematicians are interested in which natural numbers k can be expressed as a
sum of three cubes. Prior to this year, it had been established that this is
possible for all k < 1000 except for the following thirteen values:

33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975.

Earlier this year a solution for k=33 was found [1], so 42 was the next
unknown value.

[1] Brooker, A., "CRACKING THE PROBLEM WITH 33",
[https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf](https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf)

~~~
tomd3v
> Mathematicians are interested in which natural numbers k can be expressed as
> a sum of three cubes.

Why?

~~~
kadoban
Because simple questions with no known simple answers are fun to study and can
lead to actually useful techniques sometimes.

------
raghavkhanna
Here'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)

------
brokenmachine
I understand that it is interesting, but can someone ELI5 what this might be
useful for in real life?

------
densetsu
Hitchhikers rejoice!

~~~
julienchastang
I 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?

------
djyaz1200
[http://youtu.be/8190ziL5v-k](http://youtu.be/8190ziL5v-k)

------
yumraj
How was this computed? Just brute force ?

~~~
techolic
(One of) The authors previous finding was for 33, and he had a paper on that,
[https://arxiv.org/pdf/1903.04284.pdf](https://arxiv.org/pdf/1903.04284.pdf).

~~~
dagw
And even with all that optimization it still took 23 core-years to find one
solution.

------
petrikapu
In Python

    
    
      >>> (-80538738812075974)**3 + 80435758145817515**3 + 12602123297335631**3
      42

~~~
antoineMoPa
In bash

    
    
      $ echo "print((-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3)" | sed "s/\^/**/g" | python -
      42

~~~
Franciscouzo
You're just using python, using bc is more bashy:

    
    
        $ echo '(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3' | bc
        42

------
CharityEngine
Any questions about the platform they used, feel free to ask :)

------
sidcool
Any idea what algorithm was used? Was it brute forcing through all
combinations?

------
dustinmr
Now that we have the answer, I wonder what the question was?

And why is that mouse looking at me?

------
t_minus_3
dumb question: What is the practical application of this conjecture?

------
JustSomeNobody
So, does this mean life, the universe, and everything are all cubes?

~~~
asdfman123
No, it's just that Deep Thought misunderstood the question. It thought it was
solving for the sum of three cubes.

------
countryqt30
If I copy and paste this formula to Google, I get a different result

~~~
saagarjha
Google seems to be suffering from a loss in precision:
[https://www.google.com/search?hl=en&q=(%2D80538738812075974)...](https://www.google.com/search?hl=en&q=\(%2D80538738812075974\)%5E3%20%2B%2080435758145817515%5E3%20%2B%2012602123297335631%5E3)

------
arthurcolle
Was the author so excited by this finding that he replaced his academic page
with just this expression?

Very interesting finding in any case!

------
airocker
Can this be used instead of RSA?

~~~
gmiller123456
No. 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.

------
AstralStorm
And the universe turned slightly weirder again.

------
archie2
Why should I care?

------
rdiddly
It's also the answer to life, the universe and everything. Which is nice.

~~~
rawburt
It's the answer to the ultimate "question" of life, the universe, and
everything. This is an important difference. What's the question?

~~~
paulmd
"What do you get when you multiply six by nine?"

~~~
jaclaz
"I may be a sorry case, but I don't write jokes in base 13."

~~~
paulmd
The 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."

------
Scoobert
Clicked hoping for Douglas Adams, got Numberwang

~~~
julienchastang
Exactly! 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_G...](https://en.wikipedia.org/wiki/42_\(number\)#The_Hitchhiker's_Guide_to_the_Galaxy)

~~~
psv1
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.

~~~
oo0shiny
You come to HN to be cool!?

~~~
MFLoon
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.

------
sabujp
i 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

------
binaural
42 is also the answer to the universe. ;-)

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

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

~~~
defnmacro
So long and thanks for all the fish.

------
gunnihinn
As a mathematician and long-time sufferer of that Douglas Adams meme:

GIVE A FUCK?

