Mathematica Bohemica (Apr 2021)

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

  • Refik Keskin,
  • Fatih Erduvan

DOI
https://doi.org/10.21136/MB.2020.0077-19
Journal volume & issue
Vol. 146, no. 1
pp. 55 – 68

Abstract

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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 $n\geq2$ 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 $B_3=35=(55)_6.$

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