Common Terms of k-Pell and Tribonacci Numbers

Abstract

Let Tm be a Tribonacci sequence, and let the k-Pell sequence be a generalization of the Pell sequence for k ≥ 2 . The first k terms are 0, 0, ..., 0, 1, and each term after the forewords is defined by linear recurrence P (k) n = 2P (k) n−1 + P (k) n−2 + ... + P (k) n−k . We study the solution of the Diophantine equation P (k) n = Tm for the positive integer (n, k, m) with k ≥ 2. We use the lower bound for linear forms in logarithms of algebraic numbers with the theory of the continued fraction.