A friend of mine loves to trot out this oddity, and I always manage to forget it afterwards. Now that I have it in mind, I will take a minute to write it here for posterity.
Theorem: There exist , irrational, such that .
In other words, the theorem asserts that an irrational raised to an irrational may be rational. This isn’t especially interesting and, while surprising, I don’t claim to have any sort of intuition violated here. However, the proof is slippery and clever.
Proof: I will take for granted that is irrational, which is a standard first theorem in an analysis course.
Consider the quantity
There are two cases to be considered here:
is rational, so the theorem is true (with ), or
is irrational, and we can compute
which also would show that the theorem holds (with , ).
Therefore the theorem is proven either way, and we remain ignorant of whether is rational or not.