FERMAT AND MERSENNE NUMBERS IN THE k-PELL SEQUENCE

B. V. Normenyo, S. E. Rihane, A. Togbe. Fermat and Mersenne numbers in the k-Pell sequence, Mat. Stud. 56 (2021), 115–123. For an integer k ≥ 2, let (Formula presented) be the k-generalized Pell sequence, which starts with (Formula presented) (k terms) and each term afterwards is defined by the recurrence (Formula presented) for all n ≥ 2. For any positive integer n, a number of the form 2ⁿ + 1 is referred to as a Fermat number, while a number of the form 2ⁿ − 1 is referred to as a Mersenne number. The goal of this paper is to determine Fermat and Mersenne numbers which are members of the k-generalized Pell sequence. More precisely, we solve the Diophantine equation (Formula presented) in positive integers n, k, a with k ≥ 2, a ≥ 1. We prove a theorem which asserts that, if the Diophantine equation (Formula presented) has a solution (n, a, k) in positive integers n, k, a with k ≥ 2, a ≥ 1, then we must have that (n, a, k) ∈ {(1,1, k), (3, 2, k), (5, 5, 3)}. As a result of our theorem, we deduce that the number 1 is the only Mersenne number and the number 5 is the only Fermat number in the k-Pell sequence.