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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Let φ denote Euler's phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n x such that q ∤ φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.