On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

Abstract

Let ( U n ) n ∈ N (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 B and N 0 N_0 such that for any b , c ∈ Z b,c\in \mathbb {Z} with b > B b> B the equation U n − b m = c U_n - b^m = c has at most two distinct solutions ( n , m ) ∈ N 2 (n,m)\in \mathbb {N}^2 with n ≥ N 0 n\geq N_0 and m ≥ 1 m\geq 1 . Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1 T_1= T_2=1 , T 3 = 2 T_3=2 and T n = T n − 1 + T n − 2 + T n − 3 T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n ≥ 4 n\geq 4 . By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 2 N_0=2 and B = e 438 B=e^{438} . The corresponding reduction algorithm is implemented in Sage.