Yesterday my post was Cadence Academic Network Asia and it was just what it says on the can. Of course success in academia is all about publications—publish or perish. But nobody likes to do more work than necessary, so how short can your PhD thesis be? Or your publication? Who holds the record?

## Shortest PhD Thesis

If you saw the movie* A Beautiful Mind*, you know something about John Nash and the Nash Equilibrium. The problem with movies like that "based on a true story" is that you don't know which bits are true and which bits Hollywood added. I actually prefer either pure fiction, or pure biography/documentary. After that movie was made, you might have missed that Nash was killed in a car crash a couple of years ago. He and his wife were ejected from the vehicle when their taxi driver lost control. They were not wearing their seat belts. PSA: Wear your seat belt, even in a taxi.

The Nash Equilibrium was the topic of his 1950 PhD dissertation, and also for what he was awarded the Nobel Prize in Economics (yes, I know it is officially The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel). What you might not know is that his dissertation was just 26 pages long and had just two references.

Was that the shortest dissertation ever? As it happens, Kurt Gödel's dissertation was even shorter. I'm not sure exactly how long it was in German but it is 22 pages in English (in *Kurt Gödel Collected Works*). His thesis established the completeness of first-order predicate calculus. A couple of years later in 1931, he published the incompleteness theorems for which he is most well-known.

But they weren't even trying compared to David Rector, whose 1966 thesis is just nine pages long. Timothy Chow has the details on MathOverflow:

Rector's thesiscomprises a title page, an abstract page, a table of contents page, 7 pages of math, a bibliography page (8 refs.), and a biographical note page. The MIT library record's "9 leaves" exclude the title/abstract/contents, which are not numbered. Except for some trivial changes in wording in the intro, the mathematical part is indeed identical to the 4-page Topology paper, vol. 5 (1966), 343-346.The thesis occupies more space since it's manually typed; not including section titles, the 4 sections are respectively 18, 23, 42, and 36 typewritten lines

## Shortest Paper

So, a four-page paper. Is that the shortest paper?

It seems likely that there would be at least one very short paper that re-states some conjecture and gives a counterexample. The most famous conjecture like this is Goldbach's conjecture that every even number greater than two is the sum of two primes. It is one of the great unsolved problems in mathematics, universally believed to be true, but nobody has been able to prove it (or find a counterexample; any counterexample would have to be bigger than 4x10^{18} since all those even numbers have been tested). If you found a counterexample, you could write a very short paper, a single number, and fame would await you.

Anyway, there is a paper just like that, a counterexample to a conjecture. Euler had conjectured, over 200 years ago, that at least n n^{th} powers would be required to sum to an n^{th} power, for n>2. For just n=2, we know this is false (3-4-5 triangle, remember your middle school geometry). Obviously, you always have to exclude the trivial case of one power equalling itself.

Anyway, here is the complete 1966 paper:

John Conway is probably most famous to computer scientists for inventing the cellular-automaton-based game of Life, and to mathematicians for his work on the classification of finite groups. He is most famous to *me* for teaching me first-year linear algebra at Cambridge. He would race through lectures as quickly as he could so that he could tell us about something more interesting than matrix algebra in the remaining 10 or 15 minutes (such as how much honey can be held on a rotating knife, or the Doomsday algorithm for quickly calculating the day of the week for any date in your head).

But for the purpose of this blog post he is famous for publishing a paper that was just two words long (an equation counting as a word) and a couple of diagrams. The title of the paper and the authors' names don't get included in word counts.

The second author, Alexander Soifer, tells the story: Basically, the two authors, on two separate plane trips, came up with two completely different solutions. Conway then wanted to see if they could get the paper published without more than those two words (they did).

## No Words at All

A sort of spoof paper on procrastination contains no words at all (remember, title, authors and footnotes don't get included in the word count).

In case you wondered just how much of a spoof this is, the *Journal of Applied Behavior Analysis* is actually a real journal and this paper was really published in it. So the shortest paper has no words at all.

**Sign up for Sunday Brunch, the weekly Breakfast Bytes email.**