# 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 \(1 \le x \le 2015\)?