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 U₀ = 0 and U₁ = 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.

Mathematics Subject Classification (2020):

• 11B39
• 11J86

Key words:

• Lucas numbers
• b-repdigits
• logarithmic height
• reduction method