TITLE: Research work on $ax^n-by^m=1$
QUESTION [5 upvotes]: I am looking for results on the equation  $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants. 
I found literature for  $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't find anything on equation $(1)$.
The closest I got is Pillai's conjecture which is a generalization of Catalan's conjecture.
Does $ax^n-by^m=1$ has infinite solution? If it does how do we find them?
Please provide related literature/reference if possible.

REPLY [8 votes]: A lot is known about such diophantine equations of the form $F(x,y)=0$ for a polynomial $F$, including results about when the number of solutons can be infinite (this happens in very rare cases that don't include the ones you are interested in) as well as efficient methods to find the solutions. I would recommend looking at the paper "The Diophantine Equation $f(x)=g(y)$" by Y. F. Bilu and R. F. Tichy. It has an extensive list of references that cover most classical results on such equations.