Abstract. In this paper, we find all all integers c having at least two representations as a difference between a Fibonacci number and a power of 2.

Let {P n } n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1 = P 2 and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Padovan numbers.Keywords Padovan numbers · repdigits · linear forms in logarithms · reduction method Mathematics Subject Classification (2010) 11B39 · 11D45 · 11D61 · 11J86 IntroductionLet {P n } n≥0 be the sequence of Padovan numbers given by P 0 = 0, P 1 = 1, P 2 = 1 and P n+3 = P n+1 + P n for all n ≥ 0.The first few terms of this sequence are {P n } n≥0 = A repdigit is a positive integer N that has only one distinct digit when written in base 10. That is, N is of the form N = d 10 ℓ − 1 9 (1) for some positive integers d, ℓ with 1 ≤ d ≤ 9.

For an integer k ≥ 2, let {F (k) n } n 2−k be the k-generalized Fibonacci sequence which starts with 0, . . . , 0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 −dy 2 = ±1 which is a k-generalized Fibonacci number, with a couple of parametric exceptions which we completely characterise. This paper extends previous work from [17] for the case k = 2 and [16] for the case k = 3.

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.

Let (Pn) n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1 = P 2 , and P n+3 = P n+1 + Pn for all n ≥ 0. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits.