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

Abstract

AbstractLet us denote by $$F_n$$ 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 $$F_n-2^x3^y=c$$ F n - 2 x 3 y = c has more than one solution $$(n,x,y)\in {\mathbb {N}}^3$$ ( n , x , y ) ∈ N 3 with $$n>1$$ n > 1 . Moreover, in the case that $$c>0$$ c > 0 we find all integers c such that the Diophantine equation has at least three solutions and in the case that $$c<0$$ c < 0 we find all integers c such that the Diophantine equation has at least four solutions.