Can all numbers be represented by (a^n) +/- (b^m) {where n and m are >1} ?

Can all of the natural numbers be expressed by (a^n) plus or minus (b^m)? Where n and m are equal or greater than two, and are whole numbers, and a and b are whole numbers. I've noticed that there is no obvious solution for numbers of the form (2^n-2), excluding 2. ie. for 6, 14, 30, 62, etc... and I was wondering, if these numbers can't in fact be expressed in the way above, whether there are any other numbers that also cannot be expressed this way.

It'd be great if someone could:

a) provide a solution for numbers of the form (2^n-2) {although the number 30 may not necessarily disprove this theory, as the numbers could be seen as Mersenne Primes times two)

b) prove that every number can be expressed this way through theory

c) provide reasoning that the numbers (2^n-2)cannot be expressed this way

Hopefully I made myself clear. This problem's been troubling me for some time; would be great to get it off my chest.

(Sorry for posting this is number theory as well; I'm not entirely sure which category it'd fall under. Also, sorry for using a question for my title- since I'm not sure what this would be classified as, I wasn't sure what to put. If I appear to be asking more than one question, or convuluting the thread, I'm sorry again. As I said, I've been pondering this problem for quite some time without result. Be great to have some help. Thanks.)