What is the shortest paper in the history of mathematics?

Question

Now-a-days a research paper can have many pages (the paper claiming the proof of abc conjecture is of $500$ pages long!). But is there any paper in the history of mathematics which we can say as the shortest paper ever published in a well-known journal which has a remarkable impact?

Answer 1

There are many research announcements, sometimes containing few lines, so the question is not very meaningful. French Comptes rendus and old issues of AMS Bulletin have many papers of one page length or less. So they can be only compared by counting letters, and I do not think this counting is a reasnoable occupation.

But if we are talking of serious papers, with complete proofs, and containing great discoveries, then this one is hard to beat:

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. 51 N4, 1964, p. 542.

It occupies substantially less than one page, including title, author and reference list. And it is really a great discovery, and frequently cited.

Answer 2

After few searching I found this paper by L. J. Lander and T. R. Parkin on counterexample to Euler's conjecture on the sum of like powers. May be this is the shortest paper?

Answer 3

Here is a summary of my findings, after chasing down the papers mentioned in the other answers and consulting some other sources such as David Rusin's "known-math" archive. Depending on how you count length, I think there are three reasonable candidates for the shortest published paper. (Counting words is an inexact science; the counts below include the title but not the references or other metadata. A formula such as $E^2 - J$ is counted as one word.)

• 16 words, 2 diagrams, no references: J. H. Conway and A. Soifer, Covering a triangle with triangles, Amer. Math. Monthly 112 (2005), 78. This was mentioned in another answer. As Soifer explains in his book How Does One Cut a Triangle?, he and Conway submitted it to the American Mathematical Monthly as a regular article. However, one caveat is that their contribution was published not as an article in its own right, but as a "boxed filler" at the end of another article. It does not appear in the table of contents, nor was it assigned its own DOI. (The paper was later republished in Geombinatorics volume XIV, but I have not seen what the full text looks like.) Also, even though the word count is low, the two diagrams take up more space on the page than some of the other contenders.

• 56 words, no diagrams, 1 reference: L. J. Lander and T. R. Parkin, Counterexample to Euler’s conjecture on sums of like powers, Bull. Amer. Math. Soc. 72 (1966), 1079. If you think that Conway and Soifer's paper doesn't count, then this paper by Lander and Parkin, mentioned in another answer, is probably the winner. But there are some ambiguities with how to measure the "length" of a formula and a bibliographic reference. I have counted the displayed formula as a single "word" but it probably should count as more than that.

• 113 words, no diagrams, no references: E. Nelson, A proof of Liouville's theorem, Proc. Amer. Math. Soc. 12 (1961), 995. This paper occupies about the same amount of vertical space on the page as Lander and Parkin, so you could argue that it's a tie. But as far as word count goes, Nelson's paper is wordier, even if you parse the formulas in Lander and Parkin as consisting of multiple "words."

The paper by Milnor mentioned in another answer is more mathematically significant than any of the above three, but it cannot compete on the grounds of sheer brevity. But here are a couple of papers that I feel deserve an honorable mention:

• 87 words, 2 diagrams, 2 references> K. A. Dahlke, A heptomino of order 76, J. Combin. Theory Ser. A 51 (1989), 127–128. This is similar in flavor to Conway–Soifer, minus the controversy about whether it is a "real paper."

• 95 words, no diagrams, 1 reference: P. H. Doyle, Plane separation, Proc. Camb. Phil. Soc. 64 (1968), 291.

• 111 words, no diagrams, 1 reference: D. Lubell, A short proof of Sperner's lemma, J. Combin. Theory 1 (1966), 299. This paper has actually been quite influential and, as you'll see if you follow the link, has been reprinted in Gessel and Rota's anthology, Classic Papers in Combinatorics.

Answer 4

"The shortest paper ever published in a serious math journal", according to Fermat's Library:

J. H. Conway and A. Soifer, Can n² + 1 unit equilateral triangles cover an equilateral triangle of side > n, say n + ε?

Answer 5

If you see at the bottom of the contents of American Mathematical Monthly it has a section called "Mathbits".
These are papers less than a page in length.
Maybe there you can find many short papers.