On Thabit and Williams numbers base b as sum or difference of Fibonacci and Mulatu numbers and vice versa

Abstract

Let \(\left( F_n\right) _{n \ge 0}\) and \(\left( M_n\right) _{n \ge 0}\) be the Fibonacci and Mulatu sequences. Let b be a positive integer such that \(b\ge 2.\) In this paper, we prove that the following Diophantine equation

$$\begin{aligned} F_n+\epsilon M_m=\rho \left( (b\pm 1)b^{\ell }\pm 1\right) , \end{aligned}$$

where \((\epsilon , \rho )\in \{(1, 1), (-1, 1), (-1, -1)\}\) have only finitely many solutions in non-negative integers n,  m,  b and positive integer \(\ell .\) Additionally, we present a method to determine all solutions within the range \(2 \le b \le 12.\) All this is done using linear forms in logarithms of algebraic numbers and Dujella–Pethő’s reduction method.