On b-repdigits as product of consecutive of Lucas members

Abstract

A b-repdigit is a positive integer that has only one distinct digit in its base b expansion, i.e. a number of the form a(b^m – 1)=(b – 1), for some positive integers m ≥ 1, b ≥ 2 and 1 ≤ a ≤ b – 1. Let r; s be non-zero integers with r ≥ 1 and s ∈ {±1}, let {U_n } _{n ≥0} be the Lucas sequence given by U_{n +2} = rU _n+1 + sU _n , with U ₀ = 0 and U ₁ = 1: In this paper, we give effective bounds for the solutions of the Diophantine equation