# Short but Substantial Math Papers [closed]

I am looking for short papers that made a significant impact on the mathematics community. I have already seen: interesting-but-short-math-papers and, What is the Shortest Ph.D. Thesis? on math overflow, but these weren't quite what I was looking for (although the intersection of the set of answers to this question with the set of answers to either of the above links is likely to be non-trivial)

I am more interested in short and important works of mathematics, not necessarily Ph.D.s (but not necessarily not Ph.D.s either). Things that changed the course of mathematical history, that sort of thing.

Any suggested readings would be very much appreciated.

# EDIT:

I realize I was not clear on what I meant by short. around the 20 page or less mark. If it goes higher, but is really something for its size then that is acceptable too.

• A classic in short papers is John Nash's Equilibrium Points in N-Person Games with $1$ page(something which today would maybe not be considered a paper anymore). With $26$ pages a little over, but compensating this through its impact, could be Goedel's Ueber formal Unentscheidbare Saetze der Principia Mathematica und verwandter System, I.
– blub
Aug 31, 2018 at 13:14
• This MO question could also be of interest : mathoverflow.net/questions/7330/… (the first two answers there are the paper by Riemann in Barry Cipra's answer and the Nash one in the comment above). Aug 31, 2018 at 13:19
• Levin's paper on universal search problems was only 2 pages long, but introduced the idea of NP-completeness in complexity theory (which might stretch your definition of mathematics a bit). Aug 31, 2018 at 15:19
• @chepner: Why is it stretching? P=NP is still an open problem that can be written as an arithmetical sentence (namely one that only quantifies over natural numbers), and even got chosen by the Clay Mathematics Institute as 1 of only 7 Millenium Prize Problems. =) Aug 31, 2018 at 16:48
• If Riemann hypothesis is wrong and a counterexample is found, the corresponding paper would be pivotal and extremely short : "BTW, here's a non-trivial zero for Riemann zeta function which is outside the critical line : $a+bi$. kthxbye". Aug 31, 2018 at 21:40

Riemann's short paper, "Über die Anzahl der Primzahlen unter einer gegebenen Grösse," surely qualifies.

• I've made this answer community wiki so that upvotes can indicate agreement rather than reward. Aug 31, 2018 at 13:36

Not a paper, but definitely significant, is Russell's paradoxical letter to Frege.

• I've made this answer community wiki so that upvotes can indicate agreement rather than reward. Aug 31, 2018 at 13:36

I would recommend Classics of Mathematics, ed. Ronald Calinger. It's got articles from a very broad range of mathematical history, all the way from the Stone Age through 1932 (includes Gödel). Naturally, it does not include later works. Most of the most important ideas in modern mathematics will be found in here somewhere. For more modern topics, I found Love and Math by Edward Frenkel to be excellent.

Short does not mean a lot of things for me. A paper may be short, but to understand it you may have to read a lot of books. You can read the work of Milnor, he is very well-known for is concise papers which are very well written. Look for example his work on exotic spheres.

Milnor, John W. (1959), "Differentiable structures on spheres", American Journal of Mathematics, 81 (4): 962–972

There is a proof of the non-existence of vector fields on spheres by (Adams and Atiyah ?) that was famously said, could fit on a post card. There is an Indian mathematician named C.P. Ramanujan who wrote a number of very important papers in algebraic geometry and number theory. To the best of my recollection, none of his papers is over 20 pages.

The Proceedings of the AMS is all about short papers. You could probably search MathSciNet for publications in the Proceedings and sort by citation number to find some winners.

I'd do this myself, but I don't have a login right now :c

The Noah Sheets helped me a lot in contest math.

• I don't think this qualifies as an answer to the question : it seems to me that OP asks about articles that have had a lot of influence on mathematical research by introducing new ideas. Your link is just a list of known formulae. Sep 3, 2018 at 11:49
• Oh... well then I guess I can't contribute... Sep 3, 2018 at 16:08