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Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new result.
For example, I recently read and liked On the Uniqueness of the Cyclic Group of Order n (Dieter Jungnickel, The American Mathematical Monthly Vol. 99, No. 6 (Jun. - Jul., 1992), pp. 545-547, jstor, doi: 10.2307/2324062).
The paper An Empty Inverse Limit by Waterhouse is only 6 lines long, which is shorter than most abstracts. I remember finding Waterhouse's construction quite surprising and elegant when I was first learning about limits.
Ivan Niven's proof of the irrationality of $\pi$.
And Timothy Jones's longer article on the same subject that provides some intuition for Niven's proof.
Compulsory:
'On the number of primes less than a given magnitude', by Bernhard Riemann.
http://www.claymath.org/sites/default/files/ezeta.pdf
Only 10 pages, very interesting.
As a rule, you can find lots of papers of this type in the American Math Monthly. You can also find find very interesting and very short papers in the old issues of the Proceedings of the AMS (which, of course, does not mean that the recent issues do not contain papers of this type).
One of my favourite ones is a 9 lines paper by E. Nelson giving a proof of Liouville's theorem for harmonic functions; namely, that any bounded harmonic function is constant. This paper does not contain any mathematical symbol. You can find it here: A proof of Liouville's theorem.
The most surprising (to me) example of such a paper is Perelman's proof of the soul conjecture:
Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. Volume 40, Number 1 (1994), 209-212. PDF available here.
An important conjecture in differential geometry open for ~25 years, proven by Perelman in an article that's a little over 2 pages long (the actual proof is less than a page)
Don Zagier's one-sentence proof that every prime $p\equiv1$ mod $4$ is a sum of two squares, published in The American Mathematical Monthly (Vol. 97, No. 2, February, 1990, p. 144), is a gem.
My favorite short paper is Every projective variety is a quiver Grassmannian by Markus Reineke (2012). It's less than 2 pages, but a pretty cool result. John Baez wrote a nice blog explaining the result.
There's been more recent strengthening of this theorem too, that every projective variety is a quiver Grassmannian of any wild acyclic quiver. This result is contained in a five page paper, Quiver Grassmannians for wild acyclic quivers by Ringel (2018).
I think a ton of papers by Paul Erdos fall under this category. (But interesting is in the eye of the beholder)
One I like is An Elementary Proof of a Theorem of Johnson and Lindenstrauss by Sanjoy Dasgupta and Anupam Gupta (Random Struct. Alg., 22: 60–65 (2003)) - it is about 5 pages long, but a lot of that is due to inane typesetting.
Abstract. -- A result of Johnson and Lindenstrauss shows that a set of $n$ points in high dimensional Euclidean space can be mapped into an $O(\log n/ϵ^2)$-dimensional Euclidean space such that the distance between any two points changes by only a factor of $(1 ± ϵ)$. In this note, we prove this theorem using elementary probabilistic techniques.
A lot of the earlier coding theory (and surprisingly a lot of papers in IEEE Trans. Info Theory and its predecessors up to the mid 70s, the original Huffman code paper for example) papers also fit this bill, such as the original Golay code paper.
I think this paper is really cool. All the major players are there!
A Golden Product Identity for $e$, Robert P. Schneider, Mathematics Magazine, Vol. 87, No. 2 (2014), 132-134.
Abstract. -- We prove an infinite product representation for the constant e involving the golden ratio, the Möbius function and the Euler phi function -- prominent players in number theory.
On the Cohomology of Impossible Figures by Roger Penrose: Leonardo Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue (1992), pp. 245-247.
"The Simple Group of Order 604,801".
Unfortunately, I can't find it online.
A counter-example (the first one found?) disproving Euler's sum of powers conjecture.
$\checkmark$ Short
$\checkmark$ Easy to understand
$\checkmark$ Somewhat important
However,
$\checkmark$ Quite boring
What is persistent homology? (persistent homology is a very popular topic in applied topology in recent years).
Peter Higgs:
"BROKEN SYMMETRIES, MASSLESS PARTICLES AND GAUGE FIELDS" 1 page
"BROKEN SYMMETRIES AND THE MASSES OF GAUGE BOSONS". 1.5 pages
3000+ citations each and a Nobel prize later... jobs a good'n. Most of the important papers on symmetry breaking written by (to name but a few and in no particular order), Robert Brout, François Englert, Peter Higgs, Gerald Guralnik, C. Richard Hagen, and Tom Kibble, are of the letter variety and hence are a few pages.
Some may say this is physics and not maths, but that would be inaccurate as the issue of symmetry breaking and massless particles was a problem identified with, and solved by mathematics.
Deligne's "Theorie de Hodge I" is only six pages long. It contains the germs of the theory of mixed Hodge structures and it is certainly interesting and inspiring.
Terence Tao's paper on uncertainty principle of cyclic groups of prime order. This was a prerequisite to understanding the motivating example behind this hugely influential paper on compressed sensing. I especially like the various historical references to Lemma 1.3 of the first paper, which shows how fundamental it is.
Peter Lax's proof to a conjecture of Erdos on polynomials, published in Bulletin of AMS is less than five pages.
Luis Caffarelli's paper on regularity of elliptic and parabolic free boundaries is only 3 page long.