On A Conjecture Regarding Permutations Which Destroy Arithmetic Progressions

TL;DR

This paper proves Hegarty's conjecture that for all integers n not equal to 2, 3, 5, or 7, the cyclic group Z/nZ admits a permutation destroying all arithmetic progressions, using new constructions independent of previous bounds.

Contribution

The paper establishes the full conjecture for all n, removing the large bound n_0 and providing constructions inspired by Elkies and Swaminathan, independent of prior partial results.

Findings

Confirmed Hegarty's conjecture for all n ≠ 2,3,5,7

Developed new permutation constructions for Z/nZ

Results are independent of previous bounds and proofs

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 an arithmetic-progression destroying permutation. However $n_0\approx 1.4\times 10^{14}$ and thus resolving the conjecture in full remained out of reach of any computational techniques. However, this paper using constructions modeled after those used by Elkies and Swaminathan for the case of $\mathbb{Z}/p\mathbb{Z}$ with $p$ being prime, establish the conjecture in full. Furthermore our results do not rely on the fact that it suffices to study when $n<n_0$ and thus our results completely independent of the proof given by Hegarty and Martinsson.