Repdigits base b as products of two Lucas numbers

Abstract

Let (Ln) be the sequence of Lucas numbers defined by L 0 = 2, L 1 = 1, and Ln = L n−1 + L n−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, then LmLn ∈ {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, L 2 = 3 = (11)2, L 3 = 4 = (11)3, L 4 = 7 = (11)6 and L 4 = 7 = (111)2, L 6 = 18 = (33)5, L 6 = 18 = (22)8. © 2020, © 2020 NISC (Pty) Ltd.