On some Pillai type problems

In 2002, Mihailescu proved the famous Catalan conjecture that $3^2 - 2^3 = 1$ is the only solution to $a^n - b^m = 1$. For any $c > 1$ however, Pillai's more general conjecture that the equation $a^n - b^m = c$ has at most finitely many solutions is still open. Much more can be said if $a$ and $b$ are fixed, and there exist many different results. Some of them have been generalised to linear recurrence sequences, i.e. $a^n$ and $b^m$ have been replaced by fixed linear recurrence sequences $U_n$ and $V_m$. In this talk we consider the equation $U_n - b^m = c$, where $U_n$ is a given linear recurrence sequence and $b$ is fixed but arbitrary. Our main tool is lower bounds for linear forms in logarithms. This talk is based on joint work with Sebastian Heintze, Robert Tichy and Volker Ziegler.