Let \( \{L_n\}_{n\ge 0} \) be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation \(L_n-2^x3^y=c\) , for \(n,x,y\in \mathbb {Z}_{\ge 0}\) . We treat the cases \(c\in -\mathbb {N}\) , \(c=0\) and \(c\in \mathbb {N}\) independently. In the cases that \(c\in \mathbb {N}\) and \(c\in -\mathbb {N}\) , we find all integers c such that the Diophantine equation has at least three solutions. These cases are treated independently, since we employ quite different techniques in proving the two cases.
