Are primes differentiable?

Calculus Level 4

A function \(f_p(x) = \lfloor n+p \sin{x} \rfloor\) for \( x \in (0, \pi)\), \( n \in \mathbb{Z}\) & \(p\) is a prime number. \(N_p\) is the number of points where \(f_p(x)\) is not differentiable. Find the last three digits of \(N_{17497}\).

Fun fact:

\(17497\) is the \(2014\)th prime.

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