Numbers of the form $a^m-b^n$

Question

Can all positive integers $k$, be written as a difference of two perfect powers $k=a^m-b^n$, with $m,n>1$ and $a,b$ positive integers?

A number is imperfect if it can not, which numbers are imperfect?

What is the asymptotics of the number of imperfect numbers less then $x$, as $x\rightarrow\infty$?

I have proved that all odd numbers are the difference of two squares. How to solve the other cases?

Answer 1

See http://oeis.org/A074981 and references there.