Can all numbers be represented by (a^n) plus or minus (b^m) {where n and m are >1} ?

[karpmage]

Can all numbers be represented by (a^n) plus or minus (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 to 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. i.e. 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. Thanks.

 

[daon2]

Is this problem even known to be true?

If not, you may be asking in the wrong place.

 

[karpmage]

Is this problem even known to be true?

If not, you may be asking in the wrong place.

I'm not sure... :S I've asked a number of maths forums in the hope one of them will know. Sorry if this isn't the right place to ask.

 

[lookagain]

Can all of the natural numbers be expressed by
(a^n) plus or minus (b^m)?

[1] Where n and m are equal to or greater than two, and are whole numbers, and

[2] 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. i.e. for 6, 14, 30, 62, etc...

Edit: I misunderstood the post upon which I responded.
I deleted a certain section.

 

[lookagain]

Can all of the natural numbers be expressed by
(a^n) plus or minus (b^m)? Where n and m are equal to 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. i.e. 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.

\(\displaystyle Other \ \ (2n^2 - 2) \ \ numbers:\)


n = 9

\(\displaystyle 160 = 144 + 16 = 12^2 + 4^2 \ \ or\)

\(\displaystyle 160 = 196 - 36 = 14^2 - 6^2\)



n = 15

\(\displaystyle 448 = 484 - 36 = 22^2 - 6^2 \)