# Interesting but short math papers?

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).

• "Interesting" is extremely subjective. Interesting to whom? – Robert Israel Jan 5 '14 at 7:16
• MR1264984 (94m:03053) Alex J. Wilkie. On defining $C^\infty$, J. Symbolic Logic, 59 (1), (1994), 344. – Andrés E. Caicedo Jan 5 '14 at 7:26
• Two versions of this question on MO: 1 and 2. – Andrés E. Caicedo Jan 5 '14 at 7:29
• MR0346112 (49 #10838) Richard B. Darst. Most infinitely differentiable functions are nowhere analytic, Canad. Math. Bull., 16, (1973), 597–598. – Andrés E. Caicedo Jan 5 '14 at 16:12
• I think what you want is more like a mathematics journal or annual publication (they don't call them magazines but that's basically what they sound like they are). – DanielV May 21 '14 at 2:03

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.

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.

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.

• Well, no mathematical symbols such as "⍺" or "Ω" or ">", but one must understand the context-specific meaning of "bounded" and "harmonic". Even given that, it does paint a very clear picture of how two things separated in Euclidian space can be arbitrarily close to being the same thing for sufficiently large sizes of "thing" and sufficiently small relative values of "close". – Rob Fagen May 8 at 14:25

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)

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.

On the Cohomology of Impossible Figures by Roger Penrose: Leonardo Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue (1992), pp. 245-247.

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.

• Any details to share? – vonbrand May 20 '14 at 23:18
• 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. – Squirtle May 21 '14 at 2:20

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.

"The Simple Group of Order 604,801".

Unfortunately, I can't find it online.

• Ha, I thought this was about the Janko group. Nope, it's prime. :P – Richard D. James May 21 '14 at 4:28

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

• "A direct search on the CDC 6600 yielded". Note that the year was 1966, so to properly appreciate it, one has to recall the timeline of the computer development. Fortunately, it is just one click away: computerhistory.org/timeline/?year=1964 (I linked directly to the year relevant to the passage in the article but looking at a few adjacent years will certainly give you a better picture of what these words really meant when they were published). – fedja May 21 '14 at 21:55

What is persistent homology? (persistent homology is a very popular topic in applied topology in recent years).

Peter Higgs:

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.

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.

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.