Geodesic
Repdigits in the base $b$ as sums of four balancing numbers
Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 55-68
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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 
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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

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