
7373170279850 - henning
http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01116-3/home.html
======
Cogito
Basically, there is a problem known as Waring's Problem[1] which asks if for
any natural number _k_ we can find a positive number _s_ , such that we can
express any other natural number as the sum of at most _s_ natural numbers
raised to the power _k_.

In this specific paper we are talking about when _k=3_ , that is "Is there a
positive integer _s_ such that every natural number can be expressed as the
sum of _s_ cubes?"

A result by Dickson in 1939 showed that every integer (except 23 and 239) can
be represented by the sum of 8 non-negative cubes. This was further refined by
Linnik who showed that large enough integers can always be represented by 7
cubes (more details in the paper, I'm just summarising).

This paper provides support for the conjecture that for sufficiently large
integers you only need 4 cubes, and find a possible lower bound on what large
enough means - greater than 7373170279850.

It's relatively easy to show that 7373170279850 cannot be written as the sum
of four cubes. The hard aspect is finding such a number in the first place,
and then determining if it is the largest such number.

Their method was to find a number _N_1_ that is not _C_4_ (where _C_s_ means
it can be written as the sum of _s_ cubes), and then check every number
between _[N_1, 10.N_1]_. They chose the number 10 by simulating pseudo-cubes
sequences, which gave them confidence 10 is a good choice. If you find a
number that is _C_4_ in the interval, call this _N_2_ and repeat the process,
if you don't then you have found a candidate for the largest.

There are some more details about number theory tricks they used to reduce the
search space in the paper, but that seems to be the gist of the whole thing.

[Edited to include more information from the paper]

[0] direct link to the paper:
[http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-0...](http://www.ams.org/journals/mcom/2000-69-229/S0025-5718-99-01116-3/S0025-5718-99-01116-3.pdf)

[1]
[http://en.wikipedia.org/wiki/Waring's_problem](http://en.wikipedia.org/wiki/Waring's_problem)

~~~
weavie
> A result by Dickson in 1939 showed that every integer (except 23 and 239)
> can be represented by the sum of 8 non-negative cubes

I'm not sure I follow this. Which 8 cubes add up to 2?

~~~
neumino
1^3+1^3+0^3+0^3+0^3+0^3+0^3+0^3?

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liquidise
Number theory conjectures that can be [possibly] disproved by selective,
brute-force programming are hazardous to my sleep schedule.

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cocoflunchy
Abstract for those who like have trouble parsing the page:

    
    
      We conjecture that 7,373,170,279,850 is the largest integer 
      which cannot be expressed as the sum of four nonnegative integral cubes.

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japaget
Despite appearances, there is no paywall. To read the paper, click on "Full-
Text PDF" about 7 lines down from the title.

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waynecochran
What about 4,556,543,113,106,166,912,761,976,150?

~~~
kristopolous
in jest or in seriousness? if in seriousness, would you mind elaborating?

~~~
waynecochran
Jest. I couldn't resist.

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teeja
That's ok, but I was more intrigued recently by

25840^2 + 43776^2 = 2584043776

which led me to Numberopedia.
[https://sites.google.com/site/numberopedia/](https://sites.google.com/site/numberopedia/)

~~~
gfodor
Meh, this is cool and all but is just a coincidence of base-10 notation, not
some fundamental theorem about the integers.

~~~
msds
In fact, searching for x such that 1+4 _x_ 10^(Ceiling[Log10[x]])-4x^2 is a
perfect square gives you tons of integers that make up half of a pair that has
this property. 123288,328768 is my favorite pair, for arbitrary reasons.

------
lbsnake7
[https://www.dpmms.cam.ac.uk/~vrn20/Waring_longer.html](https://www.dpmms.cam.ac.uk/~vrn20/Waring_longer.html)

Here is a good beginners explanation for people that don't quite understand
what's going on (like me).

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opinali
Super cool! Changing my password to this number everywhere. :)

~~~
martin-adams
Me too. Huh, what are the odds of that...

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jaredandrews
Can any one who is more familiar with mathematics explain to me the context of
this paper? The authors describe some prior research into cubes and non-
negative integers. But what was motivating this research? Is it just
interesting in general while not necessarily solving a problem? What are the
implications of this discovery if any?

------
contingencies
_A group of humans united in perverse study of totally abstract formalisms had
the idea that somewhere in a theoretical world, utterly disconnected from any
known mode of physical or experiential reality, something truly obscure that,
temporal /philosophical zeitgeist permitting, may have 'always' been true may
in fact now have finally been noticed, or, on the other hand, may not... it
may just be a dream_

Meanwhile, in a flash of brilliance leading to a derived work of equal
philosophical substance, I posit that the sum of the digits of the largest
integer which cannot be expressed as the sum of four nonnegative integral
cubes is exactly 60, and that this _may_ be evidence of a relationship between
the sets of signed integral dervied multi-dimensional primitives and the sum
of their maximum component digits.

Ahh, how I wish I had studied maths.

~~~
unclebucknasty
Not to quibble, but it should be _the maths_.

~~~
contingencies
Genuinely curious as to your reasoning there. Why use a definite article when
discussing a body of knowledge or field of endeavor? Sure, people say "the
social sciences" but they also say "social science", "carpentry", "cleaning".

Therefore, if your comment was serious then I think this is a red-handed
example of the maths field being snooty. :)

~~~
unclebucknasty
I was joking. You know, he's doing this purposely rambling, intricate
(satirical) talk, for the purpose of mocking what he finds to be the
irrelevance of the subject mathematical discovery.

Then someone (presumably from the math camp) comes along and unwittingly
responds only to a tiny trivial detail, in a manner that is as meaningless as
the parent believes the discovery itself to be.

Ahem. So...I guess it's not a very good joke if it has to be explained, but I
got a kick out of it!

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chid
Is there any use to this knowledge? Does it tell us anything else other than
for this specific problem? (Just curious)

~~~
mathattack
There can be very long periods where number theory advances are no more useful
than say string theory or literary theory. Then every once in a while you hit
something big. Much of encryption rests on what was formerly ivory tower
papers on prime numbers.

This isn't to increase or diminish the importance of the accomplishment. I
actually don't know where this belongs in the annals of math.

