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This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2 , 8) and (3 , 9). For triples, we prove the following theorem: If a ∉ { 2 , 8 } is a fixed integer larger than 1, then there are only finitely many triples (a , b , c) of pairwise distinct integers larger than 1 such that (a , b , c) , (a + 1 , b + 1 , c + 1) and (a + 2 , b + 2 , c + 2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. [ABSTRACT FROM AUTHOR]

Let { U n } n ≥ 0 and { V m } m ≥ 0 be two linear recurrence sequences. We establish an asymptotic formula for the number of integers c in the range [ - x , x ] which can be represented as differences U n - V m . In particular, the density of such integers is 0. [ABSTRACT FROM AUTHOR]

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]