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7373170279850

ams.org

109 pointshenning39 comments

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...

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

Wow wow, way interesting. I added what I think your missing citation should be.

[1] https://en.wikipedia.org/wiki/Waring's_problem

Haha thanks, I was adding context to my comment and forgot to add the link in!

> 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?

1^3+1^3+0^3+0^3+0^3+0^3+0^3+0^3?

Ah no, scratch that. I forgot about negative numbers...

Edit: It seems I forgot about zero as well!

A simple thanks for adding to the signal on the site. I enjoyed reading the article AND your comment immensely.

Thanks! I was just as interested as you in the article, and figured a summary of the paper would help many who couldn't get past the abstract. Glad you enjoyed it.

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

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.

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

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

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

Jest. I couldn't resist.

Presumably testing this is hard? If so, I like it.

I for one applaud the Fermat-like comment and lack of additional information.

http://xkcd.com/356/

For one it is larger than 7373170279850.

It has just 5 different prime divisors. That's kind of cool for such a large number.

Not at all -- a "typical" number N has ln(ln(N)) distinct prime factors, and ln(ln(4,556,543,113,106,166,912,761,976,150) is about 4.15. Having 5 prime divisors is perfectly normal for a number of this size.

Now I'm wondering if someone will provide a 4 cube division of this.

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/

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

In fact, searching for x such that 1+4x10^(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.

Feels like a ProjectEuler problem.

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).

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

Me too. Huh, what are the odds of that...

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?

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.

Not to quibble, but it should be the maths.

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. :)

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!

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

We are one step closer to immortality.

If finding such a numbers is much more difficult than checking if a number satisfies the requirements then it could be used in cryptography as an alternative to finding prime numbers or generating hashes. Although I am not sure there are enough of these numbers for any arbitrary number of squares to make it practical.

Good question, and I am also curious. I don't see how this can be used for cryptography on a large scale or how it can help us in other ways.

Take Fermat's Last Theorem for example... how has that been used to improve or change life?

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.

http://ars.userfriendly.org/cartoons/?id=20070105

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