On Fibonacci and k-Pell numbers which are close to a power of 3

Abstract

For an integer k ≥ 2, let (Pn(k))n ≥ 2 - k be the k-generalized Pell sequence which starts with 0, ..., 0, 1 (k terms) and each term afterwards is defined by the linear recurrence Pn(k) = 2Pn - 1(k) + Pn - 2(k) + ... + Pn - k(k) for all n ≥ 2. The aims of this paper is to find all Fibonacci and k-Pell numbers which are close to a power of 3. Concretely, this problem is equivalent to the resolution of the equation Pn(k) = 3m + t with the condition |t| < 3m/2 for any integer t.