# 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...

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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. –  Richard 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-Alvarez 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

Golay's single page paper describing what is today known as Golay code, a perfect code of length 23. This is even used in NASA deep space missions, and is one of the only perfect codes which are not Hamming codes.

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The 1949 paper by R.C. Bose "A Note on Fisher's Inequality for Balanced Incomplete Block Designs" arguably gave birth to the linear algebra method in combinatorics which has since been used by many to solve highly non-trivial problems as discussed here: Linear Algebra Proofs in Combinatorics?

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Funny that Witt is not mentioned here. Indeed, his papers

Theorie der quadratischen Formen in beliebigen Körpern
(introducing Witt cancellation, Witt decomposition, the Hasse-Witt invariant, the Witt ring and in a little sidestep proving that every quadratic form in $\ge 5$ variables over a $\mathfrak{p}$-adic field is isotropic)

and

Zyklische Körper und Algebren der Chrakteristik $p$ vom Grad $p^n$
(introducing Witt vectors, Artin-Schreier-Witt theory and determining the structure of complete discrete valuation rings)

at 13 resp. 14 pages are among the longest he has ever published. But they both appeared in the remarkable volume 176 of Crelle's Journal, and giving their importance, I think they still make up a reasonable answer.

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MR0011027 Chern, Shiing-shen A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. of Math. (2) 45, (1944). 747–752

Quoting from Andre Weil's review: In order to understand the true nature of the Euler-Poincaré characteristic of a (differentiable) manifold, one has to consider it as a topological invariant of a fibre-space invariantly attached to the manifold, namely, of the space of tangent unit-vectors (or "tangent sphere bundle'') to the manifold. It is therefore only natural that an intrinsic proof of the Gauss-Bonnet formula (which expresses the Euler-Poincaré characteristic as the integral of a differential form invariantly attached to the Riemannian structure) should involve the consideration of that fibre-space. This is how the author proceeds here; and his proof, as he states, is merely the simplest example of a general method in the differential-geometric study of fibre-spaces, which is developed in the paper reviewed below."

The proof is truly intrinsic, as Chern did not use an isometric imbedding of a Riemannian manifold into an Euclidean space. And it is simple to follow.

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I will vote for C.Fefferman's paper "The multiplier problem for the ball" http://mate.dm.uba.ar/~hafg/inter-u-2010/fefferman.pdf, which is only about 5 pages and he solved an open problem about multipliers, and he wrote this when he was only a teenager!

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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.

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I read all 30 previous answers, and then did "search" on this page with my browser, and to my surprise I did not find Picard's name.

Picard's proof of the Picard Little Theorem certainly qualifies for this list. See, for example Littlewood's Miscellany, where he discusses the question, "Can a PhD thesis consist of one line?"

Picard's one-line proof started an enormous body of literature in XX century, beginning with Nevanlinna theory and including Hyperbolic groups.

To be sure, Picard's original paper (CR 88(1879)1024-7) is slightly longer than one line, but the proof itself (assuming the background that was well-known in 1879) is really one line, as reproduced in Littlewood:-)

A slight generalization of this is called Picard's Great Theorem, the only theorem that I know, which has the word "Great" in its standard name:-)

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Littlewood/Picard has not been mentioned in this thread, but it has appeared elsewhere on this site: tea.mathoverflow.net/discussion/946/shortest-phd-thesis and mathoverflow.net/questions/54775/… –  Gerry Myerson Jul 15 '13 at 23:16

A. Karatsuba and Yu. Ofman (1962). "Multiplication of Many-Digital Numbers by Automatic Computers". Proceedings of the USSR Academy of Sciences 145: 293–294.

Proved that multiplication of $n$-digit numbers could be done in less than quadratic time (thus disproving a conjecture by Kolmogorov) and provided the first divide-and-conquer algorithm for arithmetic.

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J.-C. Yoccoz called

Carl L. Siegel, Iteration of analytic functions, Ann. of Math. 43(2) (1942), 607–612.

a "brief but historic article". In only 6 pages (including all necessary background) Siegel gave the ﬁrst positive solution to a small denominator problem. This had been a major unsolved issue for over 60 years, and was a big thorn in the side for Poincaré. Siegel's paper is also credited with inspiring Kolmogorov to start the circle of ideas that led to KAM Theory. Buff, Henriksen, and Hubbard did not hesitate in calling it “one of the landmark papers of the twentieth century.”

Details

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Robert Aumann's "Agreeing to Disagree" paper, at 3 pages of length, is one of the most important papers in its field.

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What about Ribet's great Inventiones paper from the 70's $\textit{A modular construction of unramified }p\textit{-extensions of }\mathbf{Q}(\mu_p)$? I think it should be mentioned!

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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.

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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. –  Misha May 25 '13 at 18:37

Évariste Galois, "Mémoire sur les conditions de résolubilité des équations par radicaux". I believe it's about 18 pages, but the foundations of Galois theory are contained within the first few pages.

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I'm surprised this hasn't been mentioned yet, but Rostislav Grigorchuk's 1980 paper in which he constructs the Grigorchuk group is just under two pages:

On the Burnside problem on periodic groups, Funkts. Anal. Prilozen. 14, No 1 (1980) 53-54.

At the time, no one realized the full significance of this group, but some of the more remarkable properties are proven in the paper.

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Cooley and Tukey (re)invented the Fast Fourier Transform with a 5-page paper in Mathematics of Computation (1965).

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Erdös' 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. Erdös, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947) 292-294

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Perelman's Proof of the soul conjecture of Cheeger and Gromoll.'' J. Differential Geom. 40 (1994), no. 1, 209–212,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214455292

is, at 3 pages (plus a paragraph of remarks), a favourite of mine, although it has some pretty tough competition here.

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L. Euler Solutio problematis ad geometriam situs pertinentis, Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 128-140.

was the famous Bridges of Königsberg paper. It was the beginning of both topology and graph theory. It is translated into English in Newman's "World of Mathematics" and in Biggs, Lloyd & Wilson's "Graph Theory 1736-1936". In Opera Omnia it is 10 pages long.

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What about Atiyah's K-theory and Reality? I know it's not that short with its 20 pages, but if you see the paper, you notice that he didn't use his space very economically. He did provide the foundation of topological K-theory though.

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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, http://arxiv.org/pdf/1110.4379 .

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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)

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)

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For 1: ncbi.nlm.nih.gov/pmc/articles/PMC221287 and ncbi.nlm.nih.gov/pmc/articles/PMC300611 are, I believe, open-access. Interestingly this is the same domain as you have linked to for Gödel's paper. Paper 3 (Zermelo) can be pirated legally due to its age, or downloaded from GDZ (like all of Mathematische Annalen): gdz.sub.uni-goettingen.de . But searching on GDZ is a major hassle; fortunately somebody else did it: math.sfsu.edu/smith/Math800/Outlines/Zermelo1904.pdf –  darij grinberg Oct 30 '11 at 0:27

I know that this question was posted almost two years ago but I cannot resist suggesting

Zagier, D. Newman's short proof of the prime number theorem. Amer. Math. Monthly 104 (1997), no. 8, 705–708.

which is difficult to beat, I think.

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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 an apt literary quote.

"However contracted, that definition is the result of expanded meditation." —Herman Melville, Moby Dick

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The paper "Zum Hilbertschen Nullstellensatz" (Mathematische Annalen, vol. 102, page 520, 1930) in which Rabinowitsch (aka. Rainich) introduced his famous trick is one small page long - the body consists of just 13 lines!

The paper consists of a slick proof of the Nullstellensatz, but the usefulness of the trick of course goes beyond that, e.g. it is used to show that $GL_n$ is an affine algebraic group...

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Sorry, but isn't it obvious that $GL_n$ is an affine algebraic group? Do you mean something different? –  David Corwin Jul 8 '12 at 21:11

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...

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"Singularities of 2-spheres in 4-space" by Fox and Milnor. Ten pages which generated hundreds of papers in knot theory.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ojm/1200691730

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The paper by Ron Graham and Bruce Rothschild which gives a really short proof (involving a complicated triple induction) of van der Waerden's theorem:

R.L. Graham and B.L. Rothschild, A short proof of van der Waerden's theorem on arithmetic progressions, Proc. American Math. Soc. 42(2) 1974, 385–386.

http://www.ams.org/journals/proc/1974-042-02/S0002-9939-1974-0329917-8/S0002-9939-1974-0329917-8.pdf

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Lawvere's paper "Quantifiers and sheaves" (1970 International Congress of Mathematicians at Nice, vol. 1, pp. 329--334) was the first publication of his work with Tierney on elementary topoi. It contains an amazing amount of information in just 6 pages.

More generally, the writings of Bill Lawvere have the highest theorem/sentence ratio I've seen (though Leonid Levin comes pretty close).

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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.

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"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

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.

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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