What are some instances of theorems which are especially unique in mathematics, i.e. for which there are not many other theorems of a similar character? An example I have in mind is Pick's theorem, since it is the only theorem I have ever seen concerning geometry of polygons with vertices on a lattice.

There are three reasons I am interested in "lonely theorems" like this:

  1. It's hard to find these results. By virtue of their uniqueness, they tend to not fall within the scope of most traditional math classes. The only reason I found out about Pick's theorem was through math competitions.

  2. Related to the last remark, lonely theorems allow those who know of them to solve problems which other people cannot (which is presumably why they tend to arise on math comps), because there are generally not alternative approaches to fall back on.

  3. Sometimes, what begins life as a lonely theorem later becomes the centerpiece of an entire new branch of mathematics. An example that comes to mind here is Mobius inversion, which was initially a trick applying to arithmetic functions, but is now of great importance for lattices & incidence algebras.


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    $\begingroup$ Are you only interested in contemporary examples of 'lonely theorems,' or would you also appreciate historical examples in the spirit of point 3? $\endgroup$ – Semiclassical Jun 16 '16 at 15:10
  • $\begingroup$ @Semiclassical Historical ones are also appreciated. $\endgroup$ – Yly Jun 16 '16 at 15:12
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    $\begingroup$ I disagree that Pick is a 'lonely' theorem. It has many fruitful applications in diophantine approximations and there are a lot of similarities with eg. minkowski's theorem. $\endgroup$ – ArtW Jun 16 '16 at 15:13
  • $\begingroup$ I dk whether Wilson's Theorem , that $(p-1)!\equiv 1 \pmod p$ when $p$ is prime,would qualify. An immediate corollary is that when $p$ is prime and $p\equiv 1 \pmod 4$ then $((p-1)/2)!$ is a square root of $-1 \pmod p,$ but there are other ways to show it, and I dk know what else you can do with Wilson's Theorem. $\endgroup$ – DanielWainfleet Jun 16 '16 at 16:12
  • $\begingroup$ Another suggestion: Ptolemy's theorem relating the lengths of the sides and diagonals of a quadrilateral inscribed in a circle. I dk anything of its history but it seems to have,or may have had,some of the attributes 1 & 2. $\endgroup$ – DanielWainfleet Jun 16 '16 at 16:24

The Bieberbach conjecture is an example of a lonely result in the sense that, while it generated much interest and almost competition, its ultimate solution by de Branges pretty much closed the field. It turned out that the result does not have many applications, and is a kind of a very high-level olympiad problem.


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