Congruence preservation and polynomial functions from \(\mathbb{Z}_n\) to \(\mathbb{Z}_m\) (Q1367021)

A function from \(\{0,1,\dots,n-1\}\) to \(\mathbb{Z}_m\) is called congruence-preserving if \(a\equiv b\pmod d\) implies \(f(a)\equiv f(b)\pmod d\) for all \(d\mid m\). The number of congruence-preserving functions is evaluated. A comparison with a formula of \textit{Z. Chen} [Discrete Math. 137, 137--145 (1995; Zbl 0849.11028)] for the number of polynomials modulo \(m\) yields a sufficient and necessary condition for all congruence-preserving functions to be polynomials. In particular, if \(m=n\) then this happens if and only if \(m\) is not divisible by 8, nor by the square of any odd prime. The density of those numbers is calculated (\(=7/\pi^2\)).