$(x^n-x^m)a=(ax^m-4)y^2$ in positive integers How do I find all positive integers $(a,x,y,n,m)$  that satisfy $ a(x^{n}-x^{m}) = (ax^{m}-4) y^{2} $  and $ m\equiv n\pmod{2} $, with $ax$ odd?

Brute force finds three solutions so far:  
a=3, x=3, y=12, n=5, m=1
a=1, x=3, y=12, n=6, m=2
a=1, x=9, y=12, n=3, m=1
which are just different ways of saying that $729 - 9 = (9-4)\times12^2.$
