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good functions

Source: 2012 China TST Test 2 p6

March 20, 2012
functionmodular arithmeticalgebrafunctional equationnumber theory

Problem Statement

Given an integer n≥2n\ge 2, a function f:Z→{1,2,…,n}f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\} is called good, if for any integer k,1≤k≤n−1k,1\le k\le n-1 there exists an integer j(k)j(k) such that for every integer mm we have f(m+j(k))≡f(m+k)−f(m)(modn+1).f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. Find the number of good functions.
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