TY - JOUR
T1 - FERMAT AND MERSENNE NUMBERS IN THE k-PELL SEQUENCE
AU - Normenyo, B. V.
AU - Rihane, S. E.
AU - Togbe, A.
N1 - Publisher Copyright:
© 2021. © B. V. Normenyo, S. E. Rihane, A. Togbe.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Fermat number
KW - Fibonacci number
KW - Mersenne number
KW - k-Pell number
KW - linear form in logarithms
KW - reduction algorithm
UR - http://www.scopus.com/inward/record.url?scp=85123237427&partnerID=8YFLogxK
U2 - 10.30970/MS.56.2.115-123
DO - 10.30970/MS.56.2.115-123
M3 - Article
AN - SCOPUS:85123237427
SN - 1027-4634
VL - 56
SP - 115
EP - 123
JO - Matematychni Studii
JF - Matematychni Studii
IS - 2
ER -