# Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are homeomorphic but not diffeomorphic to the standard 7-sphere.

What do you think?

Note: If you have a contribution, then (by definition) it will be a paper worth reading so please do give a journal reference or hyperlink!

Edit: To echo Richard's comment, the emphasis here is really on short papers. However I don't want to give an arbitrary numerical bound, so just use good judgement...

• You should probably bound the length, cuz otherwise you could just pick your favorite paper of Ratner, Grothendieck, Thurston, et cetera and the importance blows everything else away. Commented Dec 1, 2009 at 1:41
• Or Gromov, "from whose sentences people have written theses" (as I have seen someone write somewhere) Commented Dec 1, 2009 at 2:31
• The award for the corresponding question for paper titles would have to go to "H = W". Meyers and Serrin, Proc. Nat. Acad, Sci. USA 51 (1964), 1055-6. Commented Jan 6, 2010 at 2:49
• It also depends on what you define a "paper". A number of fundamental results have been announced, and their proof has been sketched, in the C.R. Acad. Sci. - and all of them are four pages long. Commented Nov 9, 2013 at 14:48
• Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSR 28: 261–272 Commented Sep 27, 2014 at 20:14

Maybe the paper of R. Brauer and Fowler, K. A. (1955): "On groups of even order", Annals of Mathematic, Second Series 62: 565–583, ISSN 0003-486X, JSTOR 1970080, MR 0074414 deserves a mention since this is generally accepted as the point when it was realised the Classification of the Finite Simple Groups might be a feasible project.

I suggest Riemann's paper on the theory of abelian functions, which although over 50 pages in length, contains the topology and homology of compact topological surfaces, the Riemann (Roch) theorem, the algebraicity of compact Riemann surfaces, an independent algebraic argument for Riemann (Roch) for plane curves, derivation of the "Brill Noether" number at least for pencils, the generalized theta function, and much more. In my opinion this paper, by introducing complex analysis into the study of plane curves, gave rise to modern algebraic geometry.

Drinfeld and Simpson's B-Structures on G-Bundles and Local Triviality, Mathematical Research Letters 2, 823-829 (1995) comes in at under seven pages and has been quite important in all the work done on principal G-bundles (such as the geometric Langlands' program).

In particular, it proved the double quotient description of G-bundles on curves (for reductive G) which had previously only been proved for $$G = SL_n$$ by Beauville and Laszlo.

The paper can be found here.

I think this is the shortest paper (1 page) with the most large title in combinatorics (24 words!):

"Alexander Burstein's Lovely Combinatorial Proof of John Noonan's Beautiful Formula that the number of $$n$$-permutations that contain the Pattern $$321$$ Exactly Once Equals $$(3/n)(2n)!/((n-3)!(n+3)!)$$"

by Doron Zeilberger, https://arxiv.org/abs/1110.4379.

Faltings' article Endlichkeitssätze für abelsche Varietäten über Zahlkörpern has only 17 pages and proves the Tate and Shafarevich conjecture for abelian varieties over number fields, which implies as a corollary the Mordell conjecture.

Boyer’s vote, in A History of Mathematics (1968, p. 395):

in 1640, the young Pascal, then sixteen years old, published an Essay pour les coniques. This consisted of only a single printed page—but one of the most fruitful pages in history. It contained the proposition described by the author as mysterium hexagrammicum, which has ever since been known as Pascal’s theorem.

My vote is:

• K.A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262-266.

This historical paper triumphantly concludes a century-old quest to tabulate prime knots with ten of fewer crossings. There are two pages of text explaining the methodology (covering linkage numbers), and three pages of tables. A widely accepted 19th century result of Little, that writhe of reduced diagrams of the same knot is the same, is falsified by the discovery of the Perko pair at the bottom of page 263. In my opinion this may be the most interesting mathematics mistake of all time.

For more on this paper and on the fascinating story behind it, see Richard Elwes's lovely blog post, and what I wrote here.

Only a few pages in length, this paper introduced famous "wild embeddings", arguably the most famous being the Alexander Horned Sphere constructed here.

Alexander, J. W. (1924), "An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected", Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 10 (1): 8–10

A favourite of mine is Kobayashi and Wu's 4 page Annals paper "On holomorphic sections of certain Hermitian vector bundles" where they introduce the Bochner-method, which is nowadays used everywhere in differential geometry as an easy and effective method of proving vanishing results.

• Bochner method was introduced by Bochner in 1946-48 and used by numerous authors to prove vanishing theorems for cohomology groups of vector bundles ever since. This remark, of course, is not meant to diminish Kobayashi-Wu's 1970 paper. Commented May 25, 2013 at 18:37

It's probably not the winner, but it certainly deserves mentioning:

Tate's p-Divisible groups paper, although not exactly short at 26 pages, contains an incredible number of new ideas. Almost every single thread in $p$-adic geometry (e.g. in Scholze's work) traces back to this paper.

Besides introducing $p$-divisible groups, he does the cohomology computation that is the beginning of Ax–Sen–Tate theory and lead to the development of Faltings's almost mathematics; and he proves the first case of what is now known as the Hodge–Tate decomposition (and asks whether this holds in bigger generality). And that's not even what he considers his main theorem!

Here is my list (in no specific order):

(*) A proof of Ehrenfeucht's conjecture about infinite systems of equations in free groups and semigroups by Victor Guba:
V.S.Guba "Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems", Mathematical notes of the Academy of Sciences of the USSR, September 1986, Volume 40, 3, pp 688-690.

(*) A.A.Razborov, “Lower bounds on monotone complexity of the logical permanent”, Math. Notes USSR, 37:6 (1985), 485–493.
As Laszlo Lovasz put it in his talk "The Work of A.A.Razborov" (can be easily found on the Internet):
In an area where any step forward seemed almost hopeless (but which was at the same time a central area of theoretical computer science) his results meant that deep methods could be developed and to obtain strong lower bounds for algorithms was not impossible.

(*) Isaac Newton "The mathematical principles of natural philosophy" - in this case the (finite) length of the work does not matter, since the importance is infinite :)

Crux Mathematicorum, 15: 7 (1989), p. 208.

The paper 'You cannot hear the shape of the drum' by Gordon, Wolpert and Webb is very short considering the importance of the result and the sophistication of the methods used. It answers a question of Kac which was also posed in a short paper.

What about Selberg's 1947 paper?

• If you are referring to Selberg's elementary proof of the prime number theorem, I feel that this paper should have been longer and that he is not clear about what he is doing in the article. Commented Dec 12, 2019 at 6:13

How about Galois's letter written on the eve of his death and published by Liouville 17 years later?

• I think we should draw the line somewhere when it comes to what constitutes a paper. Otherwise we will have Archimedes's sketch on a bit of slate while he was down the taverna with the lads... Commented Oct 2, 2014 at 17:04
• @Yemon: Or when he was explaining some Roman soldiers about circles on the beach... Commented Oct 2, 2014 at 17:31
• +1 Maybe we should draw a line, but how can we not to mention the Galois's letter??? Commented Oct 7, 2014 at 20:05

I think this deserves to be mentioned here:

K. Hasegawa, Minimal models of nilmanifolds

In just 7 pages, using some deep results from rational homotopy theory and some basic Lie theory, the author establishes that the only even-dimensional nilmanifolds that are birationally equivalent to Kahler manifolds are tori.

But wait, there's more!

The author also shows that the only nilmanifolds that admit invariant symplectic structures are also tori.

I suppose the word "importance" in the equation can allow for some subjective input (some papers might be important to certain people, while to others not so important for their work).

This paper, entitled Finiteness of the number of compatibly-split subvarieties by Kumar and Mehta, is only 3 pages long:

https://arxiv.org/abs/0901.2098

For those who work with Frobenius splittings, it is an important result, one which was actually believed to be true for decades but not proven until 2009!

Not a winner but a strong candidate: https://en.wikipedia.org/wiki/Golod-Shafarevich_theorem

It is not a proper answer but...

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

• Not sure why it is downvoted. Commented Nov 2, 2018 at 19:06
• It's an annotation, not a mathematics paper. Commented Aug 28, 2020 at 23:45