# Like Modulo, Only For pi

Algebra Level 5

Let $$R(a,b)$$ be defined over the positive real numbers as the remainder when $$a$$ is divided by $$b$$; that is, if $$a = qb+r$$ where $$q$$ is an integer and $$0 \le r < b$$, then $$R(a,b) = r$$. For example, $$R(2 \sqrt{2}, \sqrt{2}) = 0$$ because $$\frac{2 \sqrt{2}}{\sqrt{2}} = 2$$ which has no remainder, while $$R(2, \sqrt{2}) = 2 - \sqrt{2}$$ because $$\frac{2}{\sqrt{2}} = 1 + \frac{2 - \sqrt{2}}{\sqrt{2}}$$.

Let $$f(x) = x^2 - 8x + 17$$. How many real values of $$x$$ satisfy $$R(f(x), x) = 0$$ over the interval $$1 \le x \le 2015$$?

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