if $ f^2(x)$ express $ (f(x))^2$, then the number of distinct integer solutions of g(x)=0 cannot exceed 1991. \[ f(x) - 3 = (x - x_1)^{r_1}(x - x_2)^{r_2} \cdots (x - x_n)^{r_n}R_1(x) \\ f(x) + 3 = (x - y_1)^{s_1}(x - y_2)^{s_2} \cdots (x - y_n)^{s_n}R_2(x)\] so \[ (x_i - y_1)^{s_1}(x_i - y_2)^{s_2} \cdots (x_i - y_n)^{s_n}R_2(x_i) = 6\] factors on left side must be ±1, ±2, ±3, ±6 , but each one shows only once. if n>0, m >0, then $ m \le 4, n \le 4, m + n \le 8$ if n*m=0 then $ n + m \le 1991$