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].

In this paper, we find all positive squarefree integers d such that the Pell equation X 2 − dY 2 = ±1 has at least two positive integer solutions (X, Y ) and (X , Y ) such that both X and X are sums of two Tribonacci numbers.

For an integer k ≥ 2, let {F (k) n } n≥0 be the k-generalized 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 find all integers c having at least two representations as a difference between a k-generalized Fibonacci number and a powers of 2 for any fixed k 4. This paper extends previous work from [9] for the case k = 2 and [6] for the case k = 3.

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In this paper, we find all positive squarefree integers d such that the Pell equation X 2 − dY 2 = ±1 has at least two positive integer solutions (X, Y ) and (X ′ , Y ′ ) such that both X and X ′ have Zeckendorf representations with at most two terms.This paper has been accepted for publication in SCIENCE CHINA Mathematics.