Binomial Thue equations, ternary equations, and power values of polynomials

We explicitly solve the equation $Ax^n-By^n=\pm1$ and, along the way, we obtain new results for a collection of equations $Ax^n-By^n=z^m$ with $m\in\{3,n\}$, where $x,y,z,A,B$, and $n$ are unknown nonzero integers such that $n\geq3$, $AB=p^\alpha q^\beta$ with nonnegative integers $\alpha$ and $\beta$ and with primes $2\leq p<q<30$. The proofs require a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.