Repdigits in the base $b$ as sums of four balancing numbers

Keskin, Refik, and Erduvan, Faticko. "Repdigits in the base $b$ as sums of four balancing numbers." Mathematica Bohemica (2021): 55-68. <http://eudml.org/doc/297037>.

@article{Keskin2021,
abstract = {The sequence of balancing numbers $(B_\{n\})$ is defined by the recurrence relation $B_\{n\}=6B_\{n-1\}-B_\{n-2\}$ for $n\ge 2$ with initial conditions $B_\{0\}=0$ and $B_\{1\}=1.$$B_\{n\}$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_\{n\}$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6) $. Namely, $B_\{2\}=6=(11)_\{5\}$ and $B_\{3\}=35=(55)_\{6\}.$},
author = {Keskin, Refik, Erduvan, Faticko},
journal = {Mathematica Bohemica},
keywords = {balancing number; repdigit; Diophantine equations; linear form in logarithms},
language = {eng},
number = {1},
pages = {55-68},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Repdigits in the base $b$ as sums of four balancing numbers},
url = {http://eudml.org/doc/297037},
year = {2021},
}

TY - JOUR
AU - Keskin, Refik
AU - Erduvan, Faticko
TI - Repdigits in the base $b$ as sums of four balancing numbers
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 55
EP - 68
AB - The sequence of balancing numbers $(B_{n})$ is defined by the recurrence relation $B_{n}=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_{0}=0$ and $B_{1}=1.$$B_{n}$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_{n}$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6) $. Namely, $B_{2}=6=(11)_{5}$ and $B_{3}=35=(55)_{6}.$
LA - eng
KW - balancing number; repdigit; Diophantine equations; linear form in logarithms
UR - http://eudml.org/doc/297037
ER -