On a Conjecture Regarding Permutations which Destroy Arithmetic Progressions

Abstract

Hegarty conjectured for $n\neq 2, 3, 5, 7$ that  $\mathbb{Z}/n\mathbb{Z}$ has a permutation which destroys all arithmetic progressions mod $n$. For $n\ge n_0$, Hegarty and Martinsson demonstrated that $\mathbb{Z}/n\mathbb{Z}$ has a permutation destroying arithmetic progressions. However $n_0\approx 1.4\times 10^{14}$ and thus resolving the conjecture in full remained out of reach of any computational techniques. Using constructions modeled after those used by Elkies and Swaminathan for the case of $\mathbb{Z}/p\mathbb{Z}$ with $p$ being prime, this paper establishes the conjecture in full. Furthermore, our results are completely independent of the proof given by Hegarty and Martinsson.