Geodesic
Repdigits in the base $b$ as sums of four balancing numbers
Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 55-68 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The sequence of balancing numbers $(B_{n})$ is defined by the recurrence relation $B_{n}=6B_{n-1}-B_{n-2}$ for $n\geq 2$ 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 $B_{3}=35=(55)_{6}.$
The sequence of balancing numbers $(B_{n})$ is defined by the recurrence relation $B_{n}=6B_{n-1}-B_{n-2}$ for $n\geq 2$ 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 $B_{3}=35=(55)_{6}.$
DOI : 10.21136/MB.2020.0077-19
Classification : 11B39, 11D61, 11J86
Keywords: balancing number; repdigit; Diophantine equations; linear form in logarithms 
@article{10_21136_MB_2020_0077_19,
     author = {Keskin, Refik and Erduvan, Fatih},
     title = {Repdigits in the base $b$ as sums of four balancing numbers},
     journal = {Mathematica Bohemica},
     pages = {55--68},
     year = {2021},
     volume = {146},
     number = {1},
     doi = {10.21136/MB.2020.0077-19},
     mrnumber = {4227311},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2020.0077-19/}
}
TY  - JOUR
AU  - Keskin, Refik
AU  - Erduvan, Fatih
TI  - Repdigits in the base $b$ as sums of four balancing numbers
JO  - Mathematica Bohemica
PY  - 2021
SP  - 55
EP  - 68
VL  - 146
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2020.0077-19/
DO  - 10.21136/MB.2020.0077-19
LA  - en
ID  - 10_21136_MB_2020_0077_19
ER  - 
%0 Journal Article
%A Keskin, Refik
%A Erduvan, Fatih
%T Repdigits in the base $b$ as sums of four balancing numbers
%J Mathematica Bohemica
%D 2021
%P 55-68
%V 146
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2020.0077-19/
%R 10.21136/MB.2020.0077-19
%G en
%F 10_21136_MB_2020_0077_19
Keskin, Refik; Erduvan, Fatih. Repdigits in the base $b$ as sums of four balancing numbers. Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 55-68. doi: 10.21136/MB.2020.0077-19

Cité par Sources :