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

Abstract

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 $B_3=35=(55)_6.$

Keywords

• balancing number
• repdigit
• diophantine equations
• linear form in logarithms