# 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

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.

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.

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))

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:-)

– user5117
Commented Jul 15, 2013 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 :-) Commented Jul 15, 2013 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/… Commented Jul 15, 2013 at 23:16
• In French, we say the Great Fermat Theorem instead of the Last. Commented Sep 6, 2017 at 18:52

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

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.

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. Commented Sep 20, 2016 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:

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.

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

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.

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.'' Commented Apr 11, 2013 at 13:32

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.

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.

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.

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... Commented Mar 25, 2018 at 4:36

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

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.

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).

• Commented Oct 30, 2011 at 0:30

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!

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

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.

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.

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? Commented Jul 8, 2012 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... Commented Jan 4, 2014 at 11:39

"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

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.

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.

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.)