Repdigits base <i>b</i> as products of two Lucas numbers

Abstract

Let (Ln) be the sequence of Lucas numbers defined by L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. Let 0 ≤ m ≤ n and b = 2, 3, 4, 5, 6, 7, 8, 9. In this study, we show that if LmLn is a repdigit in the base b and has at least two digits, thenLmLn ∈ {3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 28, 36, 54, 121, 228}.Furthermore, it is shown that if Ln is a repdigit in the base b and has at least two digits, then(n, b) = (2, 2), (3, 3), (4, 6), (4, 2), (6, 5), (6, 8).Namely,L2 = 3 = (11)2, L3 = 4 = (11)3, L4 = 7 = (11)6andL4 = 7 = (111)2, L6 = 18 = (33)5, L6 = 18 = (22)8.