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

Abstract

For an integer k ≥ 2, let (P_n^(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 P_n^(k) = 2P_{n - 1}^(k) + P_{n - 2}^(k) + ... + P_{n - 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 P_n^(k) = 3^m + t with the condition |t| < 3^m/2 for any integer t.