Integer coordinates

Finding the number of points with integer coordinates of the graph is similar to finding the number of integral solutions of $y=(x-2017)^{2017-x^2}$ .

We note that $y=(x-2017)^{2017-x^2}$ has integral solutions when $2017 - x^2 \ge 0$ . Since $\lfloor \sqrt{2017} \rfloor = 44$ , there are integral solutions for $-44 \le x \le 44$ or 89 solutions.

Also when $x=2016$ , then $y=(2016-2017)^{2017-2016^2} = -1$ and $x=2018$ , then $y=(2018-2017)^{2017-2018^2} = 1$ .

Therefore, the total number of integral solutions is $89+2=\boxed{91}$ .