MathDB

Iterates of function give distinct residues

Source: Abelkonkurransen Finale 2024, Problem 1b

March 8, 2024

number theorynumber theory proposedfunctionresiduemodular arithmetic

Problem Statement

Find all functions

f:\mathbb{Z} \to \mathbb{Z}

such that the numbers

n, f(n),f(f(n)),\dots,f^{m-1}(n)

are distinct modulo

m

for all integers

n,m

with

m>1

. (Here

f^k

is defined by

f^0(n)=n

and

f^{k+1}(n)=f(f^{k}(n))

for

k \ge 0