> Is it entirely obvious that 2^(p-1)+3^(p-2) -1 is always divisible by 2

Yes.

> and 3 where p is prime???

2^whatever is even, 3^whatever is odd, so 2^x + 3^y - 1 is even.

2^even = 1 (mod 3), 3^whatever = 0 (mod 3), so

2^even + 3^whatever - 1 is divisible by 3.

So if p is odd (not necessarily prime, and not 2), then the

divisibility will happen as you have said.

--Joshua ucker