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

Question

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...

Answer 1

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

https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude

Answer 2

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.

Answer 3

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)

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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.

Link to article

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)

Answer 4

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 https://www.jstor.org/stable/2323918

Answer 5

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.

Answer 6

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 ...

https://www.jstor.org/stable/2034412

Answer 7

Not sure how important, but certainly short.

Answer 8

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

Answer 9

Noam Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561-568.

Answer 10

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

Answer 11

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.

Answer 12

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.

Answer 13

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.

Answer 14

The one-page paper

Golay, Marcel J. E.: "Notes on Digital Coding", Proc. IRE 37, p. 657, 1949,

which introduces the Golay code.

Answer 15

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!)

Answer 16

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.

Answer 17

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,

Answer 18

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.

Answer 19

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.

Answer 20

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...

Answer 21

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.

Answer 22

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.

Answer 23

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.

Answer 24

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.

Answer 25

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

Answer 26

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.

Answer 27

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.

Answer 28

Mordell, L.J., On the rational solutions of the indeterminate equations of third and fourth degrees, Proc. Camb. Philos. Soc. 21 (1922), 179–192.

In this paper he proved the Mordell-Weil theorem for elliptic curves over $\mathbb{Q}$ (the group of rational points is finitely generated), and he stated the Mordell conjecture (curves of genus >1 over $\mathbb{Q}$ have only finitely many points), which was one of the most important open problems in mathematics until Faltings proved it in 1983.

Answer 29

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.

Answer 30

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."

Answer 31

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

https://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 Hasselblatt & Katok: The development of dynamics in the 20th century and the contribution of Jürgen Moser (a short discussion of the paper mentioned above can be found at p.17-18))

Answer 32

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.

Answer 33

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.

Answer 34

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:-)

Answer 35

Jannsen, Uwe (1992), "Motives, numerical equivalence and semi-simplicity", Inventions math. 107: 447–452.

Answer 36

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

Answer 37

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

https://doi.org/10.4310/jdg/1214455292

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

Answer 38

Pretty late to the party here but Kantorovich's "On the translocation of masses" from 1942 is two pages. It gave a radically new look on the Monge problem of optimal transportation and can be seen as the starting point of an immense body of work on optimal transport and distances in probability spaces.

Answer 39

I remain in awe of the 1-page paper which started the whole geometric quantization approach to representations (with symplectic manifolds, moment maps, prequantum bundles, polarizations). Unlike many “short proof” papers quoted here, it’s “just” an announcement — but as a string of true statements which ended up driving the field for decades, I find its importance/length stunning:

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Answer 40

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.

Answer 41

The little paper by John McKay on Graphs, singularities, and finite groups is a nice example.

Graphs, singularities, and finite groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 183--186, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.

Answer 42

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

Answer 43

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.

Answer 44

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.

Answer 45

My favorite has to be Mary Ellen Rudin, "An unshellable triangulation of a tetrahedron," Bull. Amer. Math. Soc. 64 (1958), 90–91. An simple but incredibly ingenious construction that makes topological combinatorics much more complicated than you think it is.

Answer 46

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.

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

Answer 47

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.

Answer 48

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.

Answer 49

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.

Answer 50

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!

Answer 51

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 given their importance, I think they still make up a reasonable answer.

Answer 52

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).

Answer 53

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...

Answer 54

Serre's GAGA isn't as short as some of the others, but it's still just over 40 pages (which is quite short by the standards of Serre/Grothendieck-style algebraic geometry at the time -- e.g. FAC is about 80 pages, and of course there are things like EGA...), and it's still GAGA.

Answer 55

The so called "Weil conjectures" are in the last pages of André Weil's short paper in 1949, "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society 55: 497–508. They probably were around before though.

Answer 56

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

From Ribet's website (pdf) or EuDML

Answer 57

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?

The paper is 2 pages long: https://doi.org/10.1214/aoms/1177729958

Here's a description of Bose and his work from the manuscript Linear Algebra Methods in Combinatorics by Babai and Frankl:

The affiliation listed on Bose’s paper is the Institute of Statistics, University of North Carolina. Before taking up residence in the U.S. in 1948, Bose worked at the Indian Statistical Institute in Calcutta. One of the most influential combinatorialists of the decades to come, Bose was forced to become a statistician by the lack of employment chances in mathematics in his native country. A pure mathematician hardly in disguise, he reared generations of combinatorialists. His students at Chapel Hill included D. K. Ray-Chaudhuri, a name that together with his student R. M. Wilson (so, may be a grandson of Bose?) will appear several dozen times on these pages for their far reaching extension of Bose’s method.

Answer 58

Two fundamental papers in computational complexity theory and the theory of formal languages are very short:

• Neil Immerman, Nondeterministic space is closed under complementation, SIAM Journal on Computing 17(5), 935–938, 1988 (four pages);

• Róbert Szelepcsényi, The method of forcing for nondeterministic automata, Bulletin of the EATCS 33, 96–100, 1987 (five pages).

Both papers independently prove what is now called the Immerman-Szelepcsényi theorem, i.e., that nondeterministic space complexity classes are closed under complement, and in particular that context-sensitive languages are closed under complement. The authors shared the Gödel Prize in 1995 for their result.

I’ve never read Szelepcsényi’s version, but Immerman’s is so short and sweet that I found it hard to believe at first that it actually works as a proof of such an important theorem.

Answer 59

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

https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-3/issue-2/Singularities-of-2-spheres-in-4-space-and-cobordism-of/ojm/1200691730.full

Answer 60

My favourite is the following tiny, self-contained article:

"Uniform equivalence between Banach Spaces" by Israel Aharoni & Joram Lindenstrauss, Bulletin of the American Mathematical Society, Volume 84, Number 2, March 1978, pp.281-283.

https://www.ams.org/journals/bull/1978-84-02/S0002-9904-1978-14475-9/S0002-9904-1978-14475-9.pdf

(in which the authors prove that there exist two non-isomorphic Banach spaces that are Lipschitz homeomorphic.)

Answer 61

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.

Answer 62

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.

Answer 63

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.

Answer 64

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.

Answer 65

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.

Answer 66

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.

Answer 67

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.

Answer 68

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

Answer 69

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.

Answer 70

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!

Answer 71

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 :)

Answer 72

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

Answer 73

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.

Answer 74

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

Answer 75

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!

Answer 76

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.

Answer 77

What about Selberg's 1947 paper?

Answer 78

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

Answer 79

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.