I would like to understand at least one of the several existing approaches to algebraic geometry over $\mathbb{F}_1$ (the field with one element). Is there an example of an "interesting" theorem that can be formulated purely in the language of ordinary schemes, but which can be proved using algebraic geometry over $\mathbb{F}_1$?

Of course, the interpretation of the word "interesting" is entirely up to your own taste. An example in which the theorem cannot be proved using "classical" methods would be most desirable, but examples where (one of) the theories of schemes over $\mathbb{F}_1$ gives an alternative proof of an already known result would also be very much appreciated.

[On a related note, perhaps there should be a tag "naive-question" for situations like this one.]