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The shortest paper ever published in a serious math journal explained (fermatslibrary.com)
109 points by fermatslibrary on Nov 24, 2015 | hide | past | favorite | 52 comments



This is really interesting, the full story is told here:

http://www.wfnmc.org/mc20101.pdf

During the years 2002–2004 I was visiting Princeton University with its fabulous mathematics department, a great fixture of which was a daily 3 to 4 PM coffee hour in the commons room, attended by everyone, from students to the Beautiful Mind (John F. Nash Jr.). For one such coffee hour, in February 2004, I came thinking—for the hundredth time in my life—about the network of evenly spaced parallel lines cutting a triangle into small congruent triangles. This time I dealt with equilateral triangles, and the crux of the matter was a demonstration that n2 unit triangles can cover a triangle of side n. I asked myself a question where the continuous clashes with the discrete: what if we were to enlarge the side length of the large triangle from n to n+ε, how many unit triangles will we need to cover it? This comprised a new open problem: Cover-Up Problem 1. Find the minimum number of unit equilateral triangles required to cover an equilateral triangle of side n + ε. During the next coffee hour, I posed the problem to a few Princeton colleagues. The problem immediately excited John H. Conway, the John von Neumann Professor of Mathematics. From the commons room he went to the airport, to fly to a conference. On board the airplane, John found a way (Figure 1) to do the job with just n2+2 unit triangles! (Area considerations alone show the need for at least n2 + 1 of them.) Conway shared his cover-up with me upon his return—at a coffee hour, of course. Now it was my turn to travel to a conference, and have quality time on 28 Mathematics Competitions Vol 23 No 1 2010 an airplane. What I found (Figure 2) was a totally different cover-up with the same number, n2 + 2 unit triangles! Upon my return, at a coffee hour, I shared my cover-up with John Conway. We decided to publish our results together. John suggested setting a new world record in the number of words in a paper, and submitting it to the American Mathematical Monthly. On April 28, 2004, at 11:50 AM (computers record the exact time!), I submitted our paper that included just two words, “n2 + 2 can” and our two drawings. I am compelled to reproduce our submission here in its entirety.


I read the story. So much for publishers not doing at least 80% of the work when it comes to publishing papers.


Another example of a short peer-reviewed paper (and related to Computer Science, yay!) is "Counterexample to Euler's conjecture on sums of like powers": http://fermatslibrary.com/s/counterexample-to-eulers-conject...


Hi Marat_Dukhan! By the way that paper has also been annotated on Fermat's Library: http://fermatslibrary.com/s/counterexample-to-eulers-conject... :)


Thanks! I updated the link


I would even say that paper is shorter, but I don't see an objective way to weight and measure the length of figures presented in a paper so it's kind of subjective.


It varies by publication, I've recently seen them counted as 250 and 450 characters (each), which would make the LA longer.


Wasn't there a single page PhD thesis that was a single sentence describing a counter-example for some long-standing conjecture?

I realize it's vague (to put it mildly), but it was 20-something years since it came up in the math class in the Uni.

Edit - Haha, found it! It was in Littlewood's Miscellany - Picard's Theorem [1].

[1] https://books.google.ch/books?id=MjVgeT7Laf8C&pg=PA40&lpg=PA...


I'm personally a fan of John Nash's paper, "Equilibrium Points in n-Person Games" [1]. It's the bottom of page 48 and the top of page 49. It's quite high on the impact to word count ratio.

My understanding (which I haven't verified) is that this is the main paper resulting in his Nobel prize.

[1] http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf


Is there a way to get rid of the 'Sign up with Facebook/Google' overlay that blocks the bottom part of the comments?


Also the "join our newsletter" popup is really disruptive.


    $(".comment-form").remove()


IIRC from their newsletter, they said that they are working on a new UI.


Right click on it, inspect element, press delete.


There's a very clear hide link to click, there's no need to open Dev tools like the other comments are suggesting.


You may be the Chosen One. I don't see a hide link on the overlay that obscures comments.


I see no such link to hide it. The only links are "Sign Up With Facebook" and "Sign Up With Google."


I refuse to click such links on principle; if they want to hide their content, I will direct my eyeballs elsewhere.


Checked in IE 11, Chrome, and FF, and I don't see any hide link.


One of the authors, John Conway, is the inventor of the Game of Life: https://en.wikipedia.org/wiki/John_Horton_Conway.


If you like GOL, check out 'golly', the game-of-life simulator. It has tons of amazing examples included, including a cpu emulator. On sourceforge and probably in your favourite OS repos, too.

The banner on their homepage is a [low res version of a] valid game-of-life set: http://golly.sourceforge.net/


His `On Numbers and Games' (https://en.wikipedia.org/wiki/On_Numbers_and_Games) is also worth a look.


Is there an answer to the question? Is an n^2 + 1 tiling possible? Is there a proof that it is not?


It looks like proving that n^2 + 1 is impossible is still an open problem: http://www.wfnmc.org/mc20101.pdf.


what is this? I really don't understand how figure 2 has to do with figure 1, what the question is or what the construction is of the narrative I'm supposed to follow...


The comments on the left explain it (I didn't understand it either). Basically, it's known that an equilateral triangle of side length n needs n^2 unit triangles to perfectly tile it (e.g. you need 16 unit triangles to tile an equilateral triangle of side length 4).

Now, if you make the side length of the large triangle just a teeny bit longer (that's what epsilon is), what is the minimum number of unit length triangles it takes to cover the larger triangle? The two different figures show two ways this can be done with n^2 + 2 triangles: the first figure essentially adds 2 triangles to the base row (the comments show how they overlap a little bit), while the second figure uses 3 overlapping triangles to make up the "tip", instead of just 1 (again, the comments show how this works because the base row doesn't need any additional triangles).

Thus, these two examples show how it can be done with n^2 + 2 triangles. It's still an open question if it can be done with n^2 + 1.


(the comments show how they overlap a little bit)

OH. Thank you. This is what I was missing; I understood "cover" to mean "tile" and thought the bottom row was just misprinted slightly.


The two figures illustrate two different ways to cover an equilateral triangle with sides slightly larger than n with n^2+2 unit size equilateral triangles. Other than that, there is no link between the two.

As mathematics papers go, this is easy to comprehend (incredibly so) but I can see that people who aren't used to the culture in mathematical literature of leaving out the obvious (http://math.stackexchange.com/questions/151782/when-is-somet...) and the trivial (https://en.m.wikipedia.org/wiki/Triviality_(mathematics)#Tri...)


This website is the TeX/Web combination I ever dreamt of!


Asher Peres has a pretty famous short paper in quantum information: http://arxiv.org/abs/quant-ph/0310035


Unrelated to the published paper. Fermatslibrary is fascinating. Is there something similar for CS papers? I would love this for some distributed systems papers.


Here's something similar for CompSci: http://paperswelove.org/

They had a presentatino on the Paxos paper about a month ago: http://www.meetup.com/Papers-We-Love-London/events/225736762...


Fermat's Library is not only for math, it has a few CS papers already, for instance http://fermatslibrary.com/s/bitcoin. If there is a particular paper you would like to be annotated, you can suggest it and maybe you'll get lucky :)


Some shorter proofs are available in the book "Proofs without Words."

But then again, that book begins with a description that visual arguments aren't truly proofs.


As one of my favorite math professors once pointed out, a proof is merely an argument that convinces people, so whatever works works.


Before the advent of computers we had two notions of proof in mathematics, the French school and the Russian school.

The Russian school is basically `something that convinces humans'. The French school is all about formality.

Since computer became a thing, the field has blossomed. Look at zero knowledge proofs and various forms of interactive proof systems as examples.


But the formality is, at bottom, just a set of rules for convincing somebody.


Some people would say that the formal proof stands on its own. No need for anyone to appreciate it.


This simultaneously proves that there is a lower bound on the number of words in an academic paper. :)


Another short paper published in the American Mathematical Monthly by Doron Zeilberger can be found here: http://i.imgur.com/PSigYcb.png


It would have benefited from being a little longer. It takes about as long to figure out that they leave open the question whether n^2+1 can, as it would to come up with the same or similar constructions on your own!


I count 14 words (11 in the title, 1 in the body + 2 times "figure"), not 2. And 14 more if you count the authors names and affiliation as words.


The statement about the two words only concerns the body, and the formula n² + 2 counts for one word.


Check out THIS word then:

(x = (-b±sqrt(b^2-4 a c))/(2 a))


Why would you count a compound expression, with internal spaces, as one word?


Technically "n^2+1" has no spaces.

And I guess one way to think about it is that terms in an expression can be viewed as letters, so the expression then becomes a word.


How do you pronounce that word?


"n squared plus one"


The spaces are mere typesetting. n^2+2 is the same expression typeset a different way.


"mere typesetting" is a rather offensive way to describe a document typeset in TeX, a system invented to free mathematics from "mere typesetting".


Hey, I've used LaTex. It deserves aspersions as much as accolades :3


In the "story" PDF linked above by elbigbad it was said the title was added by the journal without input from the authors.




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