HMMT 2017 A10

Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f: \mathbb{N} \rightarrow\{0,1, \ldots, 16\}$ such that

$$f(x+17)=f(x) \text { and } f\left(x^{2}\right) \equiv f(x)^{2}+15 (\bmod 17)$$

for all integers $x \geq 1$.