On the Diophantine equation U_n - b^m = c.

Let (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B and N_0 such that for any b,c\in \mathbb {Z} with b> B the equation U_n - b^m = c has at most two distinct solutions (n,m)\in \mathbb {N}^2 with n\geq N_0 and m\geq 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T_1= T_2=1, T_3=2 and T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n\geq 4. By means of the LLL-algorithm and continued fraction reduction we are able to prove N_0=2 and B=e^{438}. The corresponding reduction algorithm is implemented in Sage. [ABSTRACT FROM AUTHOR]