  Let $ \{T_n\}_{n\geq 0} $ be the sequence of Tribonacci numbers. In this
paper, we study the exponential Diophantine equation $T_n-2^x3^y=c$, for
$n,x,y\in \mathbb{Z}_{\ge0}$. In particular, we show that there is no integer
$c$ with at least six representations of the form $T_n-2^x3^y$.
