For an integer k ≥ 2, let (F (k) n )n be the k−Fibonacci sequence which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms.In this paper, we search for powers of 2 which are sums of two k−Fibonacci numbers.The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of [3] and [6].

For an integer k ≥ 2, let (L (k) n )n be the k−generalized Lucas sequence which starts with 0, . . . , 0, 2, 1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all the integers that appear in different generalized Lucas sequences; i.e., we study the Diophantine equation L (k) n = L (ℓ) m in nonnegative integers n, k, m, ℓ with k, ℓ ≥ 2. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is a continuation of the earlier work [4].Keywords and phrases. Generalized Fibonacci and Lucas numbers, lower bounds for nonzero linear forms in logarithms of algebraic numbers, reduction method.

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Abstract. The Tribonacci sequence {Tn} n≥0 resembles the Fibonacci sequence in that it starts with the values 0, 1, 1, and each term afterwards is the sum of the preceding three terms. In this paper, we find all integers c having at least two representations as a difference between a Tribonacci number and a power of 2. This paper continues the previous work [5].