On the Wagstaff prime numbers in k-Fibonacci sequences

A Wagstaff prime is a prime number that can be written in a special exponential form involving powers of two. For any integer greater than or equal to two, the so-called $k$-generalized Fibonacci sequence is a linear sequence in which each term is obtained by adding together the preceding $k$ terms, beginning with a fixed set of initial values. In this paper, we prove that the number three is the only Wagstaff prime that appears in any of these generalized Fibonacci sequences. Our proof makes use of lower bounds for linear forms in logarithms of algebraic numbers and a refined version of the Baker–Davenport reduction method, originally developed by Dujella and Pethő.