On b-repdigit polygonal numbers

We prove a finiteness theorem concerning repdigits in base b ≥ 2 represented by a fixed quadratic polynomial. We also show that there is a finite number of polygonal numbers that are also b-repdigits for all b ≥ 2 provided that $(b,s) \notin \left\{\left(\frac{8(s-2)}{(s-4)^{2}}d+1,s\right):s \in [3,13]-\{4\}\right\}$, where s ≥ 3 denotes the number of sides of the polygon and d ∈ {1,2,...,b-1}. We illustrate this result by finding all triangular, pentagonal and heptagonal numbers that are also b-repdigits for b ∈ [2,9]. This paper is motivated by a previous work of Kafle, Luca, and Togbé who considered the same finiteness problem for b = 10 to find all pentagonal and heptagonal numbers that are also repdigits.