On the problem of Pillai with Padovan numbers and powers of 3

Let $\{P_n\_{n \ge 0}$ be the sequence of Padovan numbers defined by $P_0 = 0, P_1 = 1, P_2 = 1$, and $P_{n+3} = P_{n+1} + P_n$ for all $n \ge 0$. In this paper, we find all integers $c$ admitting at least two representations as a difference between a Padovan number and a power of $3$.