On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers

Let (Fᵣ⁽ᵏ⁾)r≥2-k and (Lᵣ⁽ᵏ⁾)r≥2-k be generalizations of the Fibonacci and Lucas sequences, where k≥2. For these sequences the initial k terms are 0, 0, … , 0, 1 and 0, 0, … , 2, 1, and each subsequent term is the sum of the preceding k terms. In this paper, we determined all first and second kinds of Thabit numbers that can be expressed as the sums of k-Fibonacci and k-Lucas numbers. We employed the theory of linear forms in logarithms of algebraic numbers and a reduction method based on the continued fraction.