6. Equation
$$
2^{m} \times 3^{n}-3^{n+1}+2^{m}=13
$$

The non-negative integer solution $(m, n)=$ $\qquad$
6. $(3,0),(2,2)$.

The original equation can be transformed into
$$
\begin{array}{l}
\left(2^{m}-3\right)\left(3^{n}+1\right)=10 \\
\Rightarrow\left\{\begin{array}{l}
3^{n}+1=2,5,10 ; \\
2^{m}-3=5,2,1 .
\end{array}\right. \\
\Rightarrow(m, n)=(3,0),(2,2) .
\end{array}
$$