Repdigits in Narayana's Cows Sequence and their Consequences
Jhon J. Bravo
Departamento de Matemáticas
Universidad del Cauca
Popayán, Calle 5 No. 4-70
Department of Pure Mathematics
University of Waterloo
200 University Ave. W.
Waterloo, ON N2L 3G1
Facultad de Ciencias en Física y Matemáticas
Universidad Autónoma de Chiapas
Blvd. Belisario Domínguez km. 1081
Tuxtla, C.P. 29050
Narayana's cows sequence satisfies the third-order
linear recurrence relation Nn =
Nn−1 + Nn−3
for n ≥ 3 with initial conditions
N0 = 0 and N1 = N2 = 1.
In this paper, we
study b-repdigits that are sums of two Narayana numbers.
As an illustration, we
explicitly determine these numbers for the bases 2 ≤ b ≤ 100.
We also obtain results on the existence of Mersenne
prime numbers, 10-repdigits, and numbers with distinct blocks of digits in
the Narayana sequence. The proof of our main theorem uses lower bounds
for linear forms in logarithms, and a version of the Baker-Davenport
reduction method in diophantine approximation.
Full version: pdf,
(Concerned with sequences
Received March 20 2020; revised versions received April 14 2020;
October 11 2020.
Published in Journal of Integer Sequences,
October 12 2020.
Journal of Integer Sequences home page