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(bm – 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 {Un } n ≥0 be the Lucas sequence given by Un +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 where a; b; n; k and m are positive integers such that 1 ≤ a ≤ b – 1; n; k ≥ 1 and 2 ≤ b ≤ 10. Then, we explicitly solve the above Diophantine equation for the Fibonacci, Pell and balancing sequences.