4.  There are $\qquad$ pairs of integers $(m, n)$ satisfying $m^{2}-11 m n-8 n^{2}=88$.
$$
4 m^{2}-44 m n-32 n^{2}=88 \times 4 \Rightarrow(2 m-11 n)^{2}=153 n^{2}+85 \times 4+12 \Rightarrow(2 m-11 n)^{2} \equiv 12(\bmod 17),
$$

and $k^{2}$ modulo 17 has remainders of $0,1,4,9,16,8,2,15,13$, so $(2 m-11 n)^{2} \equiv 12(\bmod 17)$ has no solution, hence the number of integer solutions $(m, n)$ satisfying $m^{2}-11 m n-8 n^{2}=88$ is 0.