Exponential and regular Diophantines?

Question

I am looking for a reference on connections between exponential and "regular" (polynomial) Diophantine equations. For example, I was wondering about the Catalan-Mihailescu problem and I thought of the following two problems which can perhaps be called inverses of each other:

1. Find all integers $x,y$ such that $x^2-y^3=1$.

2. Find all integers $x,y$ such that $3^x-2^y=1$.

Is there any connection known between such inverse Diophantines?

Another example is Fermat's problem of solving $x^3-y^2=2$ in integers.

1. Find all integers $x,y$ such that $x^3-y^2=2$.

2. Find all integers $x,y$ such that $3^x-5^y=2$.

Thanks for any help.