# 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

## 79 Answers

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.

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.

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.

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:-)

• About your last sentence: Theorema Egregium comes very close... – user5117 Jul 15 '13 at 17:31
• As I understand "Egregium" was the name given by Gauss himself. In the case of Picard, it was centainly given by OTHERS :-) – Alexandre Eremenko Jul 15 '13 at 17:51
• 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
• In French, we say the Great Fermat Theorem instead of the Last. – user56097 Sep 6 '17 at 18:52

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.

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

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

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.

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.

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.

I think S.T. Yau's paper that proves the Calabi conjecture is a good example. It's 2 pages long(!) and it got him a Fields medal(along with other works, certainly). It also contains a lot of other new results (!!)

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.

• It is actually one of the pioneering papers in linear programming in general. – Michael Greinecker Sep 20 '16 at 22:18

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:

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...

• 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
• @David: It is not completely obvious: the definition of $\operatorname{GL}_n$ expresses it as a Zariski-open subset of affine space, whereas to be affine it needs to be a Zariski-closed subset. The trick in question is to introduce an extra variable... – Pete L. Clark Jan 4 '14 at 11:39

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

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.

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.

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).

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

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.

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.

• Adams' review of this paper ends with The reviewer is conscious that the paper contains points of interest not mentioned above; he pleads that this is a paper of 19 pages which cannot be summarised adequately in less than 20, and urges topologists to read it.'' – Peter May Apr 11 '13 at 13:32

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.

• Twelve pages actually seems a bit long for this... – Igor Rivin Mar 25 '18 at 4:36

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.

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.

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.

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

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.

http://www.ams.org/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.)

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.

"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

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 .