Brilliant Number theoretical Functional Equations

Define a function \( f : \mathbb{Z} \rightarrow \{1,2,...,2017 \}\) to be brilliant if for any integer \( n \) where \( 1 \leq n \leq 2016 \) there exists a certain integer \( p(n) \) so that we have:

\( f(m + p(n)) \equiv f(m +n) - f(m) \pmod{2018} \)

Find the number of brilliant functions.

Details and assumptions - 2017 is prime. - This is not an original problem. This has been modified from a competition problem whose source I have forgotten.

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