Repdigits in the base b as sums of four balancing numbers

The sequence of balancing numbers (Bn) is defined by the recurrence relation Bn = 6Bn−1 −Bn−2 for n > 2 with initial conditions B0 = 0 and B1 = 1. Bn is called the nth 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 Bn is repdigit in the base b and has at least two digits, then (n, b) = (2, 5), (3, 6). Namely, B2 = 6 = (11)5 and B3 = 35 = (55)6.