Interesting but short math papers?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.

Solution 5:

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)