Does each positive integer can be represent in difference of power sum? Let $a,b,m,n$ are positive integers with, if $m= 1$ then $n\ne 1$ and vice versa.
Let define 
$S_m(a)=1^m+2^m+...+a^m$
and $S_n(b)=1^n+2^n+...+b^n$

Question

Show that every $k\in \mathbb{N}$ can be represented as
$$ k = S_m(a)-S_n(b)$$


Example
$1=1^2+2^2-(1+2)$
$2= 1+2-(1^2)$
$3= 1^4+2^4-(1^2+2^2+3^2)$
$4= 1^2+2^2-(1^2)$
$5= 1^2+2^2+3^2-(1^3+2^3)$
$6= 1^3+2^3+3^3-(1^2+2^2+3^2+4^2)$
$7= 1^4+2^4-(1+2+3+4)$
$8= 1^4+2^4-(1^3+2^3)$
$9=1^2+2^2+3^2-(1^2+2^2)$
$10= 1+2+3+4+5-(1^2+2^2)$
$\vdots$

Probably, most positive integers $k$ cannot be represented in this way, though proving such a result is, I believe, rather out of range of current technology. Perhaps $k=163$ is the first non-representable $k$, though I didn't try very hard to verify this (actually proving this for even one fixed $k$ is likely extremely hard).
