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.
@article{10_37236_7192,
author = {Mehtaab Sawhney and David Stoner},
title = {On a conjecture regarding permutations which destroy arithmetic progressions},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/7192},
zbl = {1441.11021},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7192/}
}
TY - JOUR
AU - Mehtaab Sawhney
AU - David Stoner
TI - On a conjecture regarding permutations which destroy arithmetic progressions
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/7192/
DO - 10.37236/7192
ID - 10_37236_7192
ER -
%0 Journal Article
%A Mehtaab Sawhney
%A David Stoner
%T On a conjecture regarding permutations which destroy arithmetic progressions
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/7192/
%R 10.37236/7192
%F 10_37236_7192
Mehtaab Sawhney; David Stoner. On a conjecture regarding permutations which destroy arithmetic progressions. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/7192
