# 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. – Autumn Kent Dec 1 '09 at 1:41
• Or Gromov, "from whose sentences people have written theses" (as I have seen someone write somewhere) – Mariano Suárez-Álvarez Dec 1 '09 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. – John D. Cook Jan 6 '10 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. – Delio Mugnolo Nov 9 '13 at 14:48
• Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSR 28: 261–272 – TT_ Sep 27 '14 at 20:14

A natural choice is Riemann's "On the Number of Primes Less Than a Given Magnitude" at only 8 pages long...

• what is even further impressive about this paper is the extremely humble tone in which it is written, as opposed to many papers that get written these days (mostly in comp. sci, though, less in math) – Suvrit Sep 23 '10 at 7:40

John Nash's "Equilibrium Points in n-Person Games" (Proc. Nat. Acad. Sci. 36 (1) (1950) pp 48–49, doi:10.1073/pnas.36.1.48) is only about a page and is one of the most important papers in game theory.

• Nowadays this sort of paper would not get published at all, and would likely appear just as an answer on MathOverflow. – Andrej Bauer Nov 7 '11 at 14:26
• The exact reference is indeed: Nash, Jr.F.J., Equilibrium Points in N-Person Games", Proc. Nat. Acad. Sci. U.S.A. (1950), 48-49, but if you open Adrian's link you see that it is really one page. Basically, a Nobel prize in one page. Impressive! – Valerio Capraro Mar 20 '12 at 16:29
• @Ryan Dang! Let's try that again (this should be better): pnas.org/content/36/1/48.full.pdf. – Todd Trimble Oct 24 '16 at 16:51

Paul Cohen's paper "The independence of the continuum hypothesis" in which he introduced forcing. Six pages long (and another six in the second paper, a year later) that completely changed logic and set theory.

JSTOR access (may require a paywall)

PubMedCentral (free copy)

While I'm at it, two more in set theory:

Kurt Goedel's proof of the consistency of the continuum hypothesis and the axiom of choice, a two pages long paper.

And Zermelo's paper introducing the axiom of choice, a three pages long paper proving the well ordering theorem.

Link to article (may require a paywall)

It's not a paper, and it's not groundbreaking, but it's short!

A One-Sentence Proof That Every Prime $p\equiv 1\pmod 4$ Is a Sum of Two Squares D. Zagier The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), p. 144 http://www.jstor.org/pss/2323918

• That's a beautiful one indeed. And it also has lot of potential for the (obscurity of the idea)/(line count) competition :P – Mariano Suárez-Álvarez Jan 6 '10 at 3:39
• And you can see that sentence for only $12 at JSTOR! – I. J. Kennedy May 17 '10 at 0:37 • Save 12 bucks: The involution on a finite set$S = \{(x,y,z) \in \mathbb{N}^3 : x^2 +4yz = p \} $defined by: $(x,y,z) \mapsto \left\{ \begin{array}{cc} (x+2z,z,y-x-z) & \text{if } x < y-z \\ ( 2y-x, y, y-x+z ) & \text{if } y-z < x <2y \\ ( x-2y, x-y+z, y ) & \text{if } x > 2y \end{array} \right.$ has exactly one fixed point, so$|S|$is odd and the involution defined by$(x,y,z) \mapsto (x,z,y)$also has a fixed point. – Zavosh May 21 '10 at 11:46 • One thing I've always wondered - is there any intuition behind the involution? – dvitek Sep 30 '10 at 4:16 • +1 to Mariano and drvitek. It is hardly a memorable proof. That is, unless you have some special insight or photographic memory, you're not going to remember how that involution goes. I once wrote a crabby blog post about this proof, here: topologicalmusings.wordpress.com/2008/05/04/… – Todd Trimble Nov 7 '11 at 22:09 H. Lebesgue, Sur une généralisation de l’intégrale déﬁnie, Ac. Sci. C.R. 132 (1901), 1025– 1028. The beginning of measure theory as we know it, and a very short paper. • – bezirsk Jun 26 '13 at 11:27 • I was going to link to this paper. It's one of the most beautiful papers in signal to noise ratio sense. – Gottfried William Sep 29 '14 at 13:22 I get this nominee from Halmos... E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995 9 lines long. Not the shortest paper ever, but maximizes importance/length ... http://www.jstor.org/stable/2034412 • That's amazing! I once heard this proof orally expounded by Michael Rosen, but never knew a reference until now... – David Hansen Dec 1 '09 at 15:23 • Not the shortest paper ever? Where can I find a shorter one? (With real mathematical content of course.) – Harald Hanche-Olsen Dec 1 '09 at 15:51 • William C. Waterhouse, An Empty Inverse Limit, Proceedings of the American Mathematical Society, Vol. 36, No. 2 (Dec., 1972), p. 618. The body of the paper is only 6 lines. – Autumn Kent Dec 1 '09 at 17:29 • Other shorties... from sci.math in 1994. P.H. Doyle: Plane Separation, Proc. Camb. Phil. Soc. 64 (1968) 291; MR 36#7115. H. Furstenberg: On the Infinitude of Primes, Amer. Math. Monthly 62 (1955) 353; MR 16-904. D. Lubell: A Short Proof of Sperner's Lemma, J. Comb. Theory, Ser. A, vol.1 no. 2 (1966) 299; MR 33#2558. – Gerald Edgar Dec 2 '09 at 0:21 • Nelson's an interesting character in a number of ways. math.princeton.edu/~nelson/papers/ezek.html – Michael Hardy Mar 7 '11 at 18:29 Not sure how important, but certainly short. • It's a disproof of a conjecture by no less than Euler! – David Roberts Apr 15 '15 at 2:33 • I think that "a direct search" is a little disingenuous. A fair bit of cleverness must have gone into this, especially in '66. – Igor Rivin Apr 23 '17 at 2:39 • @IgorRivin I think this is a reflection of attitudes towards computers in mathematics at the time, and the general inaccessibility of computers and computer programs of the time. – zibadawa timmy Jul 5 '17 at 1:55 • This is from Bull. Amer. Math. Soc 72.6 (1966): 1079. The AMS hosts a link at ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/… – Matt F. Feb 15 '19 at 19:10 • Is this the first conjecture disproved by the use of computers ? – Antonio Hernández Maquívar Jul 4 '19 at 14:39 Noam Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561-568. • Because its length (and simplicity), this was the first paper I ever completely read! – David Zureick-Brown Dec 1 '09 at 4:49 Kahn and Kalai's, "A counterexample to Borsuk's conjecture" is a 3-page paper which settles a sixty-year-old conjecture with an explicit counterexample in $$\mathbb{R}^{1325}$$ (and in all sufficiently high dimensions). Although the paper is 3 pages, most of that is background on the problem and references --- the construction itself is only one paragraph. They include a literary quote. "However contracted, that definition is the result of expanded meditation." —Herman Melville, Moby Dick Riemann's Habilitationsschrift, On the hypotheses which lie at the foundation of geometry, was the start of Riemannian Geometry. An English translation took up 6 pages in Nature. • That is not a paper: it's the Habilitation$\textit{lecture}$Riemann gave, which was only published posthumously. – Victor Protsak May 21 '10 at 2:19 • There is no way a paper of 6 pages in length could lay the foundations of Riemannian geometry. – Hollis Williams Dec 12 '19 at 6:05 Depending on how strict you are, this might not qualify as a paper. Hilbert's 1900 ICM talk in which he posed his 23 problems. One of the shortest papers ever published is probably John Milnor's Eigenvalues of the Laplace Operator on Certain Manifolds, Proceedings of the National Academy of Sciences of USA, 1964, p. 542 He shows that a compact Riemannian manifold is not characterized by the eigenvalues of its Laplacian. It takes him little more than half of a page. • Also on the same topic: Also Kac's Can one hear the shape of a drum? is pretty short. – Delio Mugnolo Nov 9 '13 at 14:53 Erdős' 1947 paper Some remarks on the theory of graphs'', which is just 3 pages long, gives the lower bound$R(k,k)>2^{k/2}$for the diagonal Ramsey numbers. It could have been a much shorter paper; he completes the proof of the lower bound before the end of the first page! The paper is important not just for the bound, which (essentially) hasn't been improved in 65 years, but also for the method used; although this paper wasn't the first to use the probabilistic method, it is certainly the most influential early paper to have done so. P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947) 292-294 The one-page paper Golay, Marcel J. E.: "Notes on Digital Coding", Proc. IRE 37, p. 657, 1949, which introduces the Golay code. • It's not even a whole page! – Robin Chapman Nov 2 '10 at 7:41 • I first came across Golay codes when studying electrical engineering - I recal looking up this paper, thinking I understood it but I still had to experiment with examples for several days to really believe that such a simple thing could be such a powerful error correcting code. – Selene Routley May 20 '11 at 13:01 Kazhdan's paper "On the connection of the dual space of a group with the structure of its closed subgroups" introduced property (T) and proved many of its standard properties. And it's only 3 pages long (and it contains a surprisingly large number of details for such a short paper!) The 1958 paper of Kolmogorov entitled "A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces" is four pages long. This is the paper in which he defines the entropy of a dynamical system. Endre Szemeredi's paper on the Regularity Lemma is just 3 pages long. I think that is a good candidate as well. Szemerédi, Endre (1978), "Regular partitions of graphs", Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, 260, Paris: CNRS, pp. 399–401, • Although didn't it appear beforehand in the monstrously complicated proof of Szemeredi's theorem? – Harrison Brown Dec 2 '09 at 1:36 • On that note, the proof of the Local Lemma is very short, even though the paper it appears in is 19 pages long. – Andrew D. King Oct 10 '10 at 21:06 My mention goes to V. I Lomonosov's "Invariant subspaces for the family of operators which commute with a completely continuous operator", Funct. Anal. Appl. 7 (1973) 213-214, which in less than two pages demolished numerous previous results in invariant subspace theory, many of which previously took dozens of pages to prove. It also kick-started the theory of subspaces simultaneously invariant under several operators, where it continues to be useful today. It's highly self-contained, using only the Schauder-Tychonoff theorem, if I remember correctly. • I also like Lomonosov and Rosenthal's "The simplest proof of Burnside's theorem on matrix algebras" that proves that a proper subalgebra of a matrix algebra over an algebraically closed field must have a non-trivial invariant subspace. 3 pages. – Dima Sustretov May 4 '11 at 23:40 Any of three papers dealing with primality and factoring that are between 7 and 13 pages: First place: Rivest, R.; A. Shamir; L. Adleman (1978). "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems". Communications of the ACM 21 (2): 120–126. Runner-up: P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, Proc. 35nd Annual Symposium on Foundations of Computer Science (Shafi Goldwasser, ed.), IEEE Computer Society Press (1994), 124-134. Honorable mention: Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781–793. • Also Godel, K. "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173-98 (1931) and Turing, A.M. (1936), "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2 42: 230–65 (1937) – Steve Huntsman Dec 1 '09 at 2:09 • One more FTW and in the spirit of the original question: Donaldson, S. K. Self-dual connections and the topology of smooth 4-manifolds. Bull. Amer. Math. Soc.. 8, (1983), 81–83. – Steve Huntsman Dec 1 '09 at 2:17 • Does it count if you publish in a conference proceedings with a 10-page limit (although I did buy an extra page). The full version, SIAM J. Computing 26: 1484-1509 (1997), was 26 pages. – Peter Shor Mar 19 '10 at 13:57 • I'd say it counts and then some. I never saw a bunch of military types getting all excited and nervous about a paper on abelian categories or natural proofs and trying to understand the results. – Steve Huntsman Mar 19 '10 at 14:09 Beilinson and Bernstein's paper "Localisation de$\mathfrak g$-modules" is probably the most important in geometric representation theory, and is roughly 3 pages long. • I wonder if it has ever been made available on the internet – Dima Sustretov May 4 '11 at 22:30 • It's available on the internet in the sense that I can send you a scan. – Ben Webster May 4 '11 at 23:04 • thanks! please send it then to %first name%.%second name%@gmail.com – Dima Sustretov May 4 '11 at 23:45 • Gallica link at this question. – Francois Ziegler Mar 25 '18 at 5:09 In theoretical CS, there's the Razborov-Rudich "natural proofs" paper, which weighs in at 9 pages. After introducing and defining the terminology, and proving a couple of simple lemmas, the proof of the main theorem takes only a couple of paragraphs, less than half a page if I recall correctly. Instead of answering directly about which paper (I don't know), I think that a journal with amazing importance/page ratio was Funktsional. Anal. i ego Prilozhen./Functional analysis and its applications at the time when Gel'fand was the main editor (or Kirillov at some point). Typical paper in 1970-s was of much importance, recognizable names and results nowdays, while being usually something like 4 pages. If one looks at all the volumes in 1970-s together it is just a short interval at a bookshelf, amazing compression of thousands of important results, especially in view of many junk commercial journals nowdays which flag with impact factors like the notorious Chaos, solitons and fractals... There are a very large number of very concise papers written in the USSR, back when it existed. A good example would Beilinson's paper "Coherent sheaves on$\mathbb{P}^n$and problems of linear algebra." It's probably not quite as earth-shaking as Milnor's paper, but it's also only slightly more than 1 page long. • Well, omiting all details is not the same thing as being space-efficient! :P – Mariano Suárez-Álvarez Dec 1 '09 at 2:03 • Can anyone sum up what Beilison's paper is about to me? Many thanks. – darij grinberg Jan 5 '10 at 23:36 • Darij: this paper by Eisenbud-Floystad-Schreyer expands this construction of Beilinson: arxiv.org/abs/math/0104203 (one consequence is a good algorithm which is used in practice to calculate sheaf cohomology on projective space) – Steven Sam Sep 28 '14 at 17:35 • @darijgrinberg: You've probably found this out by now, but it's basically an explicit description of the bounded derived category of coherent sheaves on$\mathbb{P}^n$. In modern terms, we'd say that Beilinson constructed (two) full exceptional collections on$\mathbb{P}^n$, by giving a "resolution of the diagonal." Of course, this is the paper that (implicitly) created those notions. It's really amazing, and well worth a read. – Daniel Litt Sep 28 '14 at 21:16 • @DanielLitt: Thank you, but I fear you overestimated my progress. I still don't know what a derived category is and don't feel that I have the time and peace of mind to read myself into them properly. – darij grinberg Sep 28 '14 at 21:38 How about Leonid Levin (1986), Average-Case Complete Problems, SIAM Journal of Computing 15: 285-286? Quite important in complexity theory, and only two pages long, although very, very dense. Robert Aumann's "Agreeing to Disagree" paper, at 3 pages of length, is one of the most important papers in its field. I would recommend one very short "paper" by Grothendieck in some IHES publications has defined algebraic de Rham cohomology. (I don't think it maximizes the ratio in question, but it is an interesting one, anyway.) BTW, it was actually part of a mail to Atiyah. It begins with 3 dots! (Maybe some private conversation was omitted). Of course, sometimes Grothendieck wrote long letters (e.g. his 700-page letter to Quillen "pursuing stacks" or his 50-page letter to Faltings on dessin d'enfant). Also, I think Grothendieck had a (short?) paper with a striking title called "Hodge conjecture is false for trivial reason", in which he pointed out that the integral Hodge conj. is not true, one has to mod out by torsion, i.e. tensored with Q. • "The general Hodge conjecture is false for trivial reasons." I'm nitpicking, but it's easily my favorite title of a math paper ever. – Harrison Brown Dec 1 '09 at 3:28 • What, you don't like "Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle"? – Qiaochu Yuan Dec 1 '09 at 19:57 • Funny as that is, it doesn't have the same oomph to it as "X is false for trivial reasons." While we're on the subject of funny titles, though, I like "Mick gets some (the odds are on his side)", which makes no sense whatsoever as a paper title until you realize what the paper's about! – Harrison Brown Dec 2 '09 at 2:01 • Correcting the title is not just nitpicking. The paper is about a generalisation of the Hodge conjecture concerning a characterisation of the filtration on rational cohomology induced by the Hodge filtration. It deals with rational cohomology and so is not concerned with the failure of the (non-generalised) Hodge conjecture for integral cohomology. The latter result is due to Atiyah-Bott. – Torsten Ekedahl Mar 19 '10 at 5:24 • My favorite title is P. Cartier, Comment l'hypothese de Riemann ne fut pas prouvee [How the Riemann hypothesis was not proved], Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), pp. 35-48, Progr Math 22, Birkhauser, Boston, 1982, MR 85f:11035. – Gerry Myerson Sep 23 '10 at 6:42 I'm torn between Tate, J. Endomorphisms of Abelian Varieties over Finite Fields, Invent Math 2, 1966, p. 134-144 Lubin, Jonathan; Tate, John. Formal complex multiplication in local fields. Ann. of Math. (2) 81 1965 380--387. and Drinfelʹd, V. G. Coverings of$p$-adic symmetric domains. (Russian) Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29--40. bearing in mind that, as I recall, the English translation is only 7 pages long. Longer than some of those above, perhaps; but maybe they win on "importance." Here are two and a half papers in homotopy theory: 1. Dan Kan introduced Kan complexes and the Kan complex approximation functor$\mathrm{Ex}^\infty$in the three-page 1956 PNAS paper "Abstract Homotopy III" (here is a JSTOR link). I can't resist pointing out his 1958 Trans. Amer. Math Soc. paper "Adjoint Functors"—clearly too long for this contest at 36 pages—where he defines an adjunction of functors on the first page. Here is a link. 2. The 1966 Quart. J. Math. Oxford paper$K$-theory and the Hopf invariant by Adams and Atiyah is only 8 pages long. I don't have a link to the paper, but here is a MathSciNet link. Adams and Atiyah use the Adams operations in$K\$-theory to solve the Hopf invariant one problem. Adams' original proof (using secondary operations) takes 85 pages—of course that paper was extraordinarily fecund in homotopy theory.

Jürgen Moser (1965), On the Volume Elements on a Manifold , Transactions of the American Mathematical Society, Vol. 120, No. 2 (Nov., 1965), pp. 286-294

http://www.jstor.org/stable/1994022

Besides the many powerful applications of the famous "Moser argument" (or "Moser trick"), the local version gives a very nice and elegant proof of the classical Darboux Theorem.

(For a nice summary of this and other papers by Jürgen Moser, I would recommend http://www.math.psu.edu/katok_a/pub/Moserhistory.pdf (a short discussion of the paper mentioned above can be found at p.17-18))

Barry Mazur "On Embeddings of Spheres", Bull. AMS v 65 (1959) only 5 1/2 pages. It introduced the method of infinite repetition in topology and allowed the proof the generalized Schoenflies conjecture.