Proving a language (ir)regular (standard methods have failed)
This can be proven, but you need some nontrivial tools to do it.

Start with the set S = {0,3,5,6, ...} of non-negative integers having an even number of 1's in their base-2 expansion.  

It is well-known that this set is "2-automatic"; that is, there is a finite automaton accepting exactly the base-2 expansions of elements of S.  Furthermore, it is well-known that this set is not ultimately periodic (that is, it is not true that there exists a period P such that after some point C, if x >= C is in S then so is x+P).  (If it were, then the associated Thue-Morse word 01101001... would be ultimately periodic, but it is known from Thue's 1912 paper not to contain any block repeated three times consecutively.)

Next, assume S is actually "3-automatic"; that is, there is an automaton accepting exactly the base-3 expansions of elements of S.  Then by a classic theorem of Cobham (Math. Systems. Theory 3 (1969) 186-192, this would imply that S is ultimately periodic.  But we have already seen it isn't.

You can find a lot more about these ideas in my book with Allouche, "Automatic Sequences".  Warning, though, our proof of Cobham is a bit flawed, and a corrected version by Rigo can be found online here:  http://arxiv.org/abs/0907.0624 .