PERMUTATIONS DESTROYING ARITHMETIC PROGRESSIONS IN FINITE CYCLIC GROUPS

Abstract. A permutation pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a, b, c) is a non-trivial 3-term AP in G, that is c − b = b − a and a, b, c are not all equal, then (pi(a), pi(b), pi(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Zn, for all n 6 ∈ {2, 3, 5, 7}. Here we prove, as a special case of a more general result, that such a permutation exists for all n ≥ n0, for some explcitly constructed number n0 ≈ 1.4 × 10 14. We also construct such a permutation of Zp for all primes p> 3 such that p ≡ 3 (mod 8). 1