Question

I am interested in theorems with unexpected conclusions. I don't mean an unintuitive result (like the existence of a space-filling curve), but rather a result whose conclusion seems disconnected from the hypotheses. My favorite is the following. Let $f(n)$ be the number of ways to write the nonnegative integer $n$ as a sum of powers of 2, if no power of 2 can be used more than twice. For instance, $f(6)=3$ since we can write 6 as $4+2=4+1+1=2+2+1+1$. We have $(f(0),f(1),\dots) = $ $(1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,\dots)$. The conclusion is that the numbers $f(n)/f(n+1)$ run through all the reduced positive rational numbers exactly once each. See A002487 in the On-Line Encyclopedia of Integer Sequences for more information. What are other nice examples of "unexpected conclusions"?

Answer 1

Polya's Theorem: Simple random walk on $\mathbb{Z}^d$ is recurrent for $d\leq2$ and transient for $d>2$.

There is also a nice connection between this theorem and infinite networks of resistors. It turns out that the resistance of the whole network $\mathbb{Z}^d$ (one puts a unit source in one point and takes away sinks to $\infty$) is finite iff corresppnding random walk is transient :)

Answer 2

The classical differential geometry results should definitely be mentioned here. Although it may seem not surprising for us, Gauss found his Theorema Egregium to be truly remarkable and unexpected. My favorite example is Gauss-Bonnet Theorem.

Answer 3

Eckmann-Hilton argument. I mean, WHY?

Answer 4

In the strict spirit of your question, the hypothesis that 1 = -1 has as conclusion that the moon is made of green cheese.

Answer 5

The first time I ever saw Cayley's Fundamental Theorem of Group Theory - i.e. every group is isomorphic to a group of permutations on a nonempty set - I was floored and I knew anything that contained a statement that bizarre that's true was something I wanted to do in life.