On a variant of Pillai's problem involving S-units and Fibonacci numbers

Abstract

Let us denote by $F_n$ the n-th Fibonacci number. In this paper we show that there exist at most finitely many integers c such that the exponential Diophantine equation $F_n-2^x{3}^y=c$ has more than one solution $(n,x,y)\in N^3$ with $n>1$ . Moreover, in the case that $c>0$ we find all integers c such that the Diophantine equation has at least three solutions and in the case that $c<0$ we find all integers c such that the Diophantine equation has at least four solutions.

Keywords: Baker’s method; Diophantine equations; Fibonacci numbers.