
The shortest paper ever published in a serious math journal explained - fermatslibrary
http://fermatslibrary.com/s/shortest-paper-ever-published-in-a-serious-math-journal-john-conway-alexander-soifer
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elbigbad
This is really interesting, the full story is told here:

[http://www.wfnmc.org/mc20101.pdf](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.

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

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Marat_Dukhan
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...](http://fermatslibrary.com/s/counterexample-to-eulers-conjecture-
on-sums-of-like-powers)

~~~
fermatslibrary
Hi Marat_Dukhan! By the way that paper has also been annotated on Fermat's
Library: [http://fermatslibrary.com/s/counterexample-to-eulers-
conject...](http://fermatslibrary.com/s/counterexample-to-eulers-conjecture-
on-sums-of-like-powers) :)

~~~
Marat_Dukhan
Thanks! I updated the link

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eps
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...](https://books.google.ch/books?id=MjVgeT7Laf8C&pg=PA40&lpg=PA40&dq=An+integral+function+never+0+or+1+is+a+constant&source=bl&ots=ov_dqTGKL1&sig=4yqDoeXwS_JKjcb8NqdpnHqQI-o&hl=en&sa=X&ved=0ahUKEwjv7Y66iarJAhXJaRQKHS-2AN0Q6AEIGzAA#v=onepage&f=false)

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amluto
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](http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf)

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

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

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

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

~~~
vacri
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/](http://golly.sourceforge.net/)

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

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

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kristopolous
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...

~~~
adevine
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.

~~~
colanderman
_(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.

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

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

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

~~~
noblethrasher
Here's something similar for CompSci:
[http://paperswelove.org/](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...](http://www.meetup.com/Papers-We-Love-
London/events/225736762/)

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xigency
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.

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

~~~
eru
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.

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

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

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outside1234
This simultaneously proves that there is a lower bound on the number of words
in an academic paper. :)

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

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pervycreeper
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!

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fermigier
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.

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

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

~~~
pd1
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.

~~~
harryjo
How do you pronounce that word?

~~~
pd1
"n squared plus one"

