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Determine the number of functions

Source: Turkish TST 2011 Problem 9

July 23, 2011
functionmodular arithmeticinductionnumber theory proposednumber theory

Problem Statement

Let pp be a prime, nn be a positive integer, and let Zpn\mathbb{Z}_{p^n} denote the set of congruence classes modulo pn.p^n. Determine the number of functions f:Zpn→Zpnf: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n} satisfying the condition f(a)+f(b)≡f(a+b+pab)(modpn) f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} for all a,b∈Zpn.a,b \in \mathbb{Z}_{p^n}.
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