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\title{\bf Permutations destroying arithmetic progressions \\ in finite cyclic groups}

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% the address should include the country, and does not have to include
% the street address 

\author{Peter Hegarty\\
\small Department of Mathematics\\[-0.8ex]
\small Chalmers and University of Gothenburg\\[-0.8ex] 
\small Gothenburg, Sweden\\
\small\tt hegarty@chalmers.se\\
\and
Anders Martinsson\\
\small Department of Mathematics\\[-0.8ex]
\small Chalmers and University of Gothenburg\\[-0.8ex]
\small Gothenburg, Sweden\\
\small\tt andemar@chalmers.se
}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Jun 18, 2015}{Nov 30, 2015}{Dec 11, 2015}\\
\small Mathematics Subject Classification: 11B75}

\begin{document}

\maketitle


\begin{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 $\mathbb{Z}_n$, for all 
$n \not\in \{2, \, 3, \, 5, \, 7\}$. Here we prove, as a special case of a 
more general result, that such a permutation exists for all $n \geq n_0$, for 
some explicitly constructed number $n_0 \approx 1.4 \times 10^{14}$. We also 
construct such a permutation of $\mathbb{Z}_p$ for all primes $p > 3$ such 
that $p \equiv 3 \; ({\hbox{mod $8$}})$.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Permutation; arithmetic progression; finite cyclic group
\end{abstract}

%\section{Introduction}

\section{Introduction}\label{sect:intro}

Let $G$ be an abelian group, $S$ a subset of $G$. A bijection 
$\pi: S \rightarrow S$ is said to \emph{destroy}{\footnote{The term 
\emph{avoid} was used in \cite{H}. The term \emph{destroy} was preferred in 
\cite{JS}, and we agree it captures the notion better.}} \emph{arithmetic 
progressions (APs)} if there is no triple $(a, \, b, \, c)$ of elements of 
$S$ satisfying
\par (i) $a, \, b, \, c$ are not all equal,
\par (ii) $c-b = b-a$, 
\par (iii) $\pi(c) - \pi(b) = \pi(b) - \pi(a)$. 
\\
\par This notion was introduced by the first author in \cite{H}, though 
earlier Sidorenko \cite{S} had given an example of such a permutation in the 
case $G = \mathbb{Z}$, $S = \mathbb{N}$. It should not be confused with the 
somewhat different, and probably more famous, notion of a permutation 
\emph{containing} no arithmetic progressions \cite{DEGS}. 

The most important open question from \cite{H} concerns the existence of 
AP-destroying permutations of finite cyclic groups $\mathbb{Z}_n$. Conjecture 
C of that paper asserts that such permutations exist if and only if 
$n \not\in \{2, \, 3, \, 5, \, 7\}$. In this paper we come close to resolving 
this conjecture in full. Before stating our main result, we need to define an 
extension of the concept of AP-destroying permutation, in the special case of 
finite cyclic groups:

\begin{definition}
Let $s, \, t \in \mathbb{N}_0$. A permutation $\pi$ of $\mathbb{Z}_n$ is said 
to \emph{destroy $(s,\,t)$-almost APs} if there is no triple $(a, \, b, \, c)$ 
of elements of $\mathbb{Z}_n$ satisfying
\par (i) $a, \, b, \, c$ are not all equal, 
\par (ii) $a+c-2b \equiv \eta_1 \; ({\hbox{mod $n$}})$ for some 
$\eta_1 \in \{0, \, \pm 1, \, \dots, \, \pm s\}$, 
\par (iii) $\pi(a) + \pi(c) - 2\pi(b) \equiv \eta_2 \; ({\hbox{mod $n$}})$ 
for some $\eta_2 \in \{0, \, \pm 1, \, \dots, \, \pm t\}$.
\end{definition}

Hence $(s,\,t) = (0,\,0)$ is the case of an AP-destroying permutation. We 
shall prove

\begin{theorem}\label{thm:main}
(i) There exists a permutation of $\mathbb{Z}_n$ destroying arithmetic 
progressions for all $n \geq n_0$ where 
\begin{equation*}\label{eq:bignumber}
n_0 = \left( 9 \times 11 \times 16 \times 17 \times 19 
\times 23 \right)^2
\approx 1.4 \times 10^{14}.
\end{equation*}
(ii) For every $s, \, t \in \mathbb{N}_0$ there is an $n_{0}(s, \, t)$ such 
that there exists a permutation of $\mathbb{Z}_n$ destroying $(s, \, t)$-almost
APs for all $n \geq n_{0}(s, \, t)$. 
\end{theorem}

While (i) may seem to be a special case of (ii), we state it separately for 
two reasons. First and foremost, we use (i) in the proof of (ii). Secondly, we 
have in this special case tried to find the best constant $n_{0}$ which our 
method will yield. Of course, we expect that Conjecture C of \cite{H} is 
true, but some additional ideas will probably be needed to prove it in full. 
\par The idea for the proof of (i) is to construct a ``master permutation'' of 
$\mathbb{Z}_{\sqrt{n_0}}$ which is $(1, \, 2)$-almost AP-avoiding, and then 
combine this with ideas from Proposition 2.3(ii) and Lemma 3.5 of \cite{H} to 
construct AP-destroying permutations of $\mathbb{Z}_n$ for all $n \geq n_0$. 
In our proof, we break down the $(1, \, 2)$-destroyal property into a total 
of $14$ simpler ones and find, by simple computer search, permutations of $6$ 
cyclic groups of pairwise relatively prime orders satisfying different subsets 
of these simpler properties. Finally, the master permutation is obtained via 
an application of the Chinese Remainder Theorem. The proof of (ii) follows a 
similar strategy, but this time the ``master permutation'' destroys 
$(2, \, 2)$-almost APs, and it requires a more subtle application of the 
aforementioned ideas from 
\cite{H} to get the final result. The full proof of Theorem \ref{thm:main} 
is presented in Section \ref{sect:pfmain}.

The values of $n_0(s, \, t)$ arising from our proof will be extremely large. 
Though we have tried to optimise the value which our method gives for 
$n_{0}(0, \, 0)$, it remains completely impractical to attempt to complete the 
proof of Conjecture C of \cite{H} by a brute-force computer search. The main 
point of our result is that we think it removes any substantial doubt whether 
the conjecture is true. We will expand on this issue in the final section of 
the paper. However, it remains interesting to try to prove the full conjecture 
and, in particular, to try to do so without resorting to any large-scale 
computer searches. It follows from Lemma 3.5 of \cite{H} that, if we let 
$\mathcal{P}$ denote the set of those $n \in \mathbb{N}$ for which 
$\mathbb{Z}_n$ admits an AP-destroying permutation, then $\mathcal{P}$ is 
closed under multiplication. Hence, a natural strategy is to first focus on 
primes. The following result will be proven in Section \ref{sect:pfprimes}:

\begin{theorem}\label{thm:primes}
Let $p$ be a prime such that $p > 3$ and $p \equiv 3 \; ({\hbox{mod $8$}})$. 
Then there exists a permutation of $\mathbb{Z}_p$ destroying arithmetic progressions.
\end{theorem}

As we shall see, there are obvious ways one could try to tinker with this 
proof so as to make it work also for other primes. So far, however, we have 
not found any such tinkering that works. This and other outstanding issues 
will be addressed in Section \ref{sect:final}.
  
\setcounter{table}{0}
%\setcounter{equation}{0}

\section{Proof of Theorem \ref{thm:main}}\label{sect:pfmain}

We introduce some further notation. Let $s, \, t \in \mathbb{N}_0$. A 
permutation $\pi$ of $\mathbb{Z}_n$ is said to \emph{destroy the pattern 
$s \mapsto t$} if there is no triple $(a, \, b, \, c)$ of elements of 
$\mathbb{Z}_n$ satisfying
\par (i) $a, \, b, \, c$ are not all equal, 
\par (ii) $a+c-2b \equiv s \; ({\hbox{mod $n$}})$, 
\par (iii) $\pi(a) + \pi(c) - 2 \pi(b) \equiv t \; ({\hbox{mod $n$}})$.
\\
\\
Hence, $\pi$ destroys $(s, \, t)$-almost APs if and only if it destroys the 
patterns
$s^{\prime} \mapsto t^{\prime}$, for all $s^{\prime} \in [-s, \, s]$ and 
$t^{\prime} \in [-t, \, t]$. In the following assertions, $\xi^{-1}$ denotes 
the inverse of $\xi$ modulo $n$. The proofs are almost trivial:

\begin{lemma}\label{lem:normalise}
(i) Suppose $\pi: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ is a permutation 
destroying the pattern $0 \mapsto 1$ and that GCD$(t, \, n) = 1$. Then 
$\pi_{1}: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ given by 
\begin{equation*}\label{eq:zerotoone}
\pi_{1} (x) = t \pi (x)
\end{equation*}
is a permutation destroying the pattern $0 \mapsto t$. 
\\
(ii) Suppose $\pi: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ is a permutation 
destroying the pattern $1 \mapsto 1$ and that GCD$(s, \, n) =$ 
GCD$(t, \, n) = 1$. Then $\pi_2: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ given by
\begin{equation*}\label{onetoone}
\pi_{2} (x) = t \pi (s^{-1} x)
\end{equation*}
is a permutation destroying the pattern $s \mapsto t$. 
\\
(iii) If $\pi: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ is a permutation 
destroying the pattern $s \mapsto t$, then $\pi^{-1}$ destroys $t \mapsto s$. 
\end{lemma}

\par The reader is encouraged to write their own program to check the 
correctness of the data in Table \ref{tab:data}, which was obtained by 
computer search. Note that we are here identifying $\mathbb{Z}_n$ with the 
set $\{0,\,1,\,\dots,\,n-1\}$, and following standard practice in identifying 
the string $(a_0,\, a_1,\,\dots,\,a_{n-1})$ with the permutation 
$\pi: i \mapsto a_i$. 

\begin{table}[ht!]
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline 
$i$ & $n_i$ & $\pi: \mathbb{Z}_{n_i} \rightarrow \mathbb{Z}_{n_i}$ & 
${\hbox{Patterns destroyed by $\pi$,}}$ 
\\ $\;$ & $\;$ & $\;$ & ${\hbox{together with $0 \mapsto 0$}}$ \\ \hline \hline
$1$ & $9$ & $(0,\,1,\,8,\,3,\,2,\,6,\,4,\,7,\,5)$ & $0 \mapsto 2, \;\; 
-1 \mapsto -2$ \\ \hline
$2$ & $11$ & $(0,\,1,\,8,\,10,\,6,\,9,\,5,\,7,\,3,\,2,\,4)$ & $0 \mapsto -2, 
\;\; -1 \mapsto 2$ \\ \hline
$3$ & $16$ & 
$(0,\,2,\,5,\,3,\,15,\,12,\,1,\,14,\,10,\,8,\,11,\,13,\,4,\,7,\,6,\,9)$ & 
$1 \mapsto 1, \;\; -1 \mapsto -1$ \\ \hline
$4$ & $17$ & 
$(0,\,1,\,3,\,9,\,11,\,7,\,4,\,8,\,15,\,12,\,16,\,10,\,14,\,5,\,2,\,13,\,6)$ & 
$-1 \mapsto 1, \;\; 1 \mapsto -1$ \\ \hline
$5$ & $19$ & $(0,\,2,\,14,\,4,\,10,\,17,\,9,\,13,\,18,\,3,\,6,\,15,$ & 
$1 \mapsto 1, \;\; -1 \mapsto 1$ \\ 
$\;$ & $\;$ & $8,\,12,\,5,\,1,\,7,\,11,\,16)$ & $\;$ \\ \hline
$6$ & $23$ & $(0,\,1,\,4,\,3,\,21,\,22,\,2,\,11,\,12,\,7,\,8,\,5,$ & 
$0 \mapsto 1, \;\; 1 \mapsto 0,$ \\ 
$\;$ & $\;$ & $10,\,9,\,6,\,19,\,16,\,15,\,20,\,17,\,18,\,13,\,14)$ & 
$0 \mapsto -1, \;\; -1 \mapsto 0$ \\ \hline
$7$ & $25$ & 
$(0,\,2,\,5,\,1,\,3,\,9,\,13,\,20,\,10,\,15,\,23,\,4,\,21,$ & 
$1 \mapsto 1, \;\; -1 \mapsto 1$ \\
$\;$ & $\;$ & $17,\,24,\,7,\,22,\,18,\,12,\,16,\,19,\,8,\,14,\,6,\,11)$ & 
$\;$ \\ \hline
$8$ & $29$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,$ & 
$1 \mapsto 1$ \\ 
$\;$ & $\;$ & $9,\,24,\,16,\,14,\,20,\,18,\,25,\,23,\,27,\,26,\,28,\,17,$ & 
$\;$ \\
$\;$ & $\;$ & $15,\,21,\,11,\,19,\,22)$ & $\;$ \\ \hline
$9$ & $31$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,$ & 
$1 \mapsto 1$ \\
$\;$ & $\;$ & 
$9,\,11,\,14,\,20,\,27,\,23,\,25,\,24,\,26,\,29,\,28,\,30,$ & $\;$ \\
$\;$ & $\;$ & $16,\,18,\,17,\,19,\,22,\,21,\,15)$ & $\;$ \\ \hline
$10$ & $37$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,\,9,$ & 
$1 \mapsto 1$ \\ 
$\;$ & $\;$ & $11,\,14,\,18,\,15,\,17,\,21,\,24,\,22,\,32,\,31,\,35,\,30,$ & 
$\;$ \\ 
$\;$ & $\;$ & $33,\,19,\,34,\,36,\,23,\,20,\,27,\,25,\,29,\,26,\,28,\,16)$ & 
$\;$ \\ \hline
$11$ & $41$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,$ & 
$1 \mapsto 1$ \\ 
$\;$ & $\;$ & $9,\,11,\,14,\,18,\,15,\,17,\,21,\,23,\,22,\,25,\,29,\,35,$ & 
$\;$ \\
$\;$ & $\;$ & $38,\,36,\,31,\,34,\,40,\,19,\,37,\,39,\,16,\,27,\,26,\,28,$ & 
$\;$ \\
$\;$ & $\;$ & $32,\,24,\,33,\,30,\,20)$ & $\;$ \\ \hline
$12$ & $43$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,$ & 
$1 \mapsto 1$ \\
$\;$ & $\;$ & $9,\,11,\,14,\,18,\,15,\,17,\,21,\,23,\,22,\,19,\,26,\,35,$ & 
$\;$ \\
$\;$ & $\;$ & $41,\,36,\,39,\,34,\,16,\,33,\,40,\,38,\,37,\,27,\,24,\,20,$ & 
$\;$ \\
$\;$ & $\;$ & $28,\,42,\,25,\,31,\,29,\,32,\,30)$ & $\;$ \\ \hline
$13$ & $47$ & $(0,\,2,\,1,\,3,\,6,\,5,\,7,\,4,\,13,\,12,\,8,\,10,$ & 
$1 \mapsto 1$ \\
$\;$ & $\;$ & $9,\,11,\,14,\,18,\,15,\,17,\,21,\,23,\,22,\,19,\,26,\,20,$ & 
$\;$ \\
$\;$ & $\;$ & $31,\,16,\,29,\,39,\,41,\,44,\,37,\,43,\,24,\,45,\,38,\,28,$ & 
$\;$ \\
$\;$ & $\;$ & $46,\,25,\,33,\,27,\,34,\,30,\,40,\,42,\,36,\,32,\,35)$ & $\;$ 
\\ \hline
$14$ & $13$ & $(0,\,1,\,4,\,2,\,7,\,6,\,12,\,9,\,11,\,8,\,3,\,5,\,10)$ & 
$0 \mapsto 1$ \\ \hline
$15$ & $49$ & $(0,\,1,\,4,\,2,\,3,\,6,\,7,\,12,\,5,\,8,\,9,\,15,\,11,$ & 
$0 \mapsto 1$ \\
$\;$ & $\;$ & $13,\,10,\,16,\,14,\,21,\,20,\,22,\,28,\,17,\,25,\,18,\,19,$ & 
$\;$ \\
$\;$ & $\;$ & $23,\,24,\,35,\,38,\,40,\,37,\,43,\,44,\,48,\,45,\,41,\,42,$ & 
$\;$ \\
$\;$ & $\;$ & $31,\,47,\,46,\,26,\,32,\,36,\,27,\,30,\,29,\,39,\,34,\,33)$ & 
$\;$ \\ \hline
%$16$ & $53$ & $(0,\,1,\,4,\,2,\,3,\,6,\,7,\,12,\,5,\,8,\,9,\,15,\,11,\,13,$ 
%& $0 \mapsto 1$ \\
%$\;$ & $\;$ & 
%$10,\,16,\,14,\,21,\,20,\,22,\,28,\,17,\,25,\,18,\,19,\,23,\,24,$ & $\;$ \\
%$\;$ & $\;$ & 
%$34,\,32,\,33,\,26,\,43,\,41,\,48,\,27,\,44,\,47,\,45,\,46,\,50,$ & $\;$ \\
%$\;$ & $\;$ & 
%$49,\,36,\,51,\,29,\,37,\,38,\,52,\,30,\,31,\,42,\,40,\,35,\,39)$ & $\;$ 
%\\ \hline
\end{tabular} 
\end{center}
\vspace{0.3cm}
\caption{Permutation data}
\label{tab:data}
\end{table}%$\;$
For each $i \in \{1,\,2,\,3,\,4,\,6\}$, let $\pi_i$ be the permutation of 
$\mathbb{Z}_{n_i}$ given in Table \ref{tab:data}. Let $\pi_5$ be the 
permutation of $\mathbb{Z}_{19}$ given by $\pi_{5}(x) = 2 \pi^{-1}(x)$, where 
$\pi$ is as in Table \ref{tab:data}, and observe that, by 
Lemma \ref{lem:normalise}, $\pi_{5}$ destroys the patterns 
$0 \mapsto 0$, $1 \mapsto 2$, $1 \mapsto -2$. Thus, for each of the $14$ 
non-zero pairs 
$(s_i, \, t_i) \in \{-1,\, 0, \, 1\} \times \{-2, \, -1, \, 0, \, 1, \, 2\}$, 
there is some $i \in [1, \, 6]$ such that $\pi_i$ destroys the pattern 
$s_i \mapsto t_i$. Let 
$\sigma: \mathbb{Z}_{\sqrt{n_0}} \rightarrow \prod_{i=1}^{6} \mathbb{Z}_{n_i}$ be 
the natural isomorphism of abelian groups given by the Chinese Remainder 
Theorem, i.e.: 
$\sigma( x \; ({\hbox{mod $\sqrt{n_0}$}})) = 
\prod_{i=1}^{6} (x \; ({\hbox{mod $n_i$}}))$. We claim that the map 
$\pi_0: \mathbb{Z}_{\sqrt{n_0}} \rightarrow \mathbb{Z}_{\sqrt{n_0}}$ given by 
\begin{equation}\label{eq:master}
\pi_0 = \sigma^{-1} \circ (\pi_1, \, \dots, \, \pi_{6}) \circ \sigma
\end{equation}
is a permutation destroying $(1,\, 2)$-almost APs. This will be our master 
permutation for the proof of Theorem \ref{thm:main}(i).

We now show that $\pi_0$ has the desired property. Let $s \in \{0, \pm 1\}$ 
and let $(a, b, c)$ be a non-trivial (that is, $a, b, c$ are not all equal) 
triple of elements in $\mathbb{Z}_{\sqrt{n_0}}$ such that 
$a+c-2b \equiv s\;(\hbox{mod $\sqrt{n_0}$})$. Let 
$(a_i, \, b_i, c_i)$, $i = 1,\,\dots,\,6$, be the projections on the various 
factors of this triple, after applying $\sigma$. We note that 
$a_i + c_i - 2b_i \equiv s \; ({\hbox{mod $n_i$}})$, and
\begin{equation}\label{eq:modmaster}
\pi_0(a) + \pi_0(c)-2\pi_0(b) \equiv 
\pi_i(a_i) + \pi_i(c_i)-2\pi_i(b_i)\,(\hbox{mod $n_i$})
\end{equation}
for every $i$. Since $(a, b, c)$ is non-trivial, there must be at least one 
factor, $i_1$ say, such that $(a_{i_1}, b_{i_1}, c_{i_1})$ is non-trivial. We 
consider two cases:
\\
\\
{\sc Case 1:} There is some $i_2$ such that $a_{i_2}=b_{i_2}=c_{i_2}$. 
\\
\\
Clearly, this can only occur if $s=0$, so $(a, b, c)$ and all its projections 
are APs. As $\pi_{i_1}$ is AP-destroying, we have 
$\pi_{i_1}(a_{i_1})+\pi_{i_1}(c_{i_1})-2 \pi_{i_1}(b_{i_1}) 
\not\equiv 0\;(\hbox{mod $n_{i_1}$})$. Furthermore, we trivially have 
$\pi_{i_2}(a_{i_2})+\pi_{i_2}(c_{i_2})-2 \pi_{i_2}(b_{i_2}) \equiv 
0\;(\hbox{mod $n_{i_2}$})$. Hence, by (\ref{eq:modmaster}), 
$\pi_0(a) + \pi_0(c)-2\pi_0(b)$ is a non-zero multiple of $n_{i_2}>2$.
\\
\\
{\sc Case 2:} $(a_i, \, b_i, \, c_i)$ is non-trivial for every $i$.
\\
\\
For any $t \in \{0, \pm 1, \pm 2\}$ we have that, by choice of 
$\pi_1, \dots \pi_6$, there exists an $i$ such that $\pi_i$ destroys the 
pattern $s\mapsto t$. Hence, we have 
$\pi_0(a)+\pi_0(c)-2\pi_0(b) \not \equiv t\;(\hbox{mod $\sqrt{n_0}$})$ as, by 
\ref{eq:modmaster}, they are not even congruent modulo $n_i$.
\\
\\
This completes the proof that $\pi_0$ destroys (1, 2)-almost APs.
\\
\par To prove Theorem \ref{thm:main}(i), it thus remains to show how to use 
the master permutation $\pi_0$ to construct an AP-destroying permutation of 
$\mathbb{Z}_n$ for every $n \geq n_0$. To begin with, let $m, \, n$ be any 
positive integers and write $n = k \cdot m + l$, where $0 \leq l < m$. Place 
the numbers $0,\,1,\,\dots,\,n-1$ clockwise around a circle, and divide them 
up into consecutive blocks $B_0,\,\dots,\,B_{m - 1}$, each of which has size 
$k$ or $k+1$. Thus there will be exactly $l$ blocks of size $k+1$. Let 
$\beta(x)$ denote the number of the block containing $x$, i.e.: 
$x \in B_{\beta(x)}$. We make two claims:
\\
\\
{\sc Claim 1:} If $k \geq m$ then no matter which blocks have size $k+1$, if 
$(a, \, b, \, c)$ is an AP modulo $n$, then 
$\beta(a) + \beta(c) - 2\beta(b) \in \{0, \, \pm 1, \, \pm 2\} \; 
({\hbox{mod $m$}})$. 
\\
\\
To see this, consider a ``worst case'' where $a = 0$ and $b$ is the furthest 
clockwise (last) element of block $B_i$, for some $0 \leq i < m/2$ and such 
that $2b < n$. Then $k(i+1)-1 \leq b \leq (k+1)(i+1) - 1$ and so 
$2k(i+1) - 2 \leq 2b \leq 2(k+1)(i+1) - 2$. For the claim to hold, we need 
$2b$ to lie in one of the blocks $B_{2i-2}, \, \dots, \, B_{2i+2}$. The last 
element of $B_{2i-3}$ is at most $(k+1)(2i-2)$, while the first element of 
$B_{2i+3}$ is at least $k(2i+3)$. Hence the claim holds provided
\begin{equation*}\label{eq:ineqs}
(k+1)(2i-2) < 2k(i+1) - 2 \;\;\;\; {\hbox{and}} \;\;\;\;
2(k+1)(i+1)-2 < k(2i+3).
\end{equation*} 
Both inequalities are easily checked to hold provided $k \geq m$. The 
symmetric ``worst case'' where $a$ is the last and $b$ the first element in 
their respective blocks is handled similarly.
\\
\\
{\sc Claim 2:} For any $n$, if numbers are placed in blocks according to 
$\beta(x) := \lfloor mx/n \rfloor$, then every block has size $k$ or $k+1$ 
and, for any $(a, \, b, \, c)$ an AP modulo $n$, one has the stronger 
conclusion that 
$\beta(a) + \beta(c) - 2\beta(b) \in \{0, \, \pm 1\} \; ({\hbox{mod $m$}})$. 
\\
\\
This claim is easily verified by plugging in the formula for $\beta(x)$. 
\\
\par Now suppose $n \geq n_0$. Write $n = k \cdot \sqrt{n_0} + l$, where 
$0 \leq l < \sqrt{n_0}$. Imagine the numbers $0,\,1,\,\dots,\,n-1$ placed 
clockwise around a circle. We shall describe a rearrangement of this circular 
string of $n$ numbers such that, if $\pi(x)$ denotes the location of the 
number $x$ after the rearrangement, then $\pi$ will be an AP-destroying 
permutation of $\mathbb{Z}_n$. 

\par Firstly, divide the $n$ numbers into consecutive clockwise blocks 
$B_0,\,\dots,\,B_{\sqrt{n_0} - 1}$, such that 
$\beta(x) = \lfloor \frac{x \sqrt{n_0}}{n} \rfloor$. For each 
$i = 0,\,1,\,\dots,\,\sqrt{n_0} - 1$, let $\tau_i$ be a permutation of the 
elements of block $B_i$ which destroys APs, considering the elements of the 
block as lying in the group $\mathbb{Z}$ of ordinary integers. It follows from 
Proposition 2.3(ii) of \cite{H} that such permutations exist. Note that the 
$\tau_i$ will automatically also destroy APs modulo $n$. Our AP-destroying 
permutation $\pi$ of $\mathbb{Z}_n$ is gotten by first rearranging the blocks 
according to the master permutation $\pi_0$, and then applying $\tau_i$ within 
each block (or vice versa, the two operations commute). In other words, after 
applying $\pi$ to the circular arrangement of numbers, the blocks 
$B_{\pi_{0}^{-1}(0)}, \, B_{\pi_{0}^{-1}(1)}, \, \dots, \, B_{\pi_{0}^{-1}(\sqrt{n_0}-1)}$ 
appear in clockwise order and, within block $B_i$, its integer elements have 
been permuted according to $\tau_i$. Since $\pi_0$ is $(1,\,2)$-almost 
AP-destroying, it is easily deduced from Claims 1 and 2 that $\pi$ destroys 
APs modulo $n$. This completes the proof of Theorem \ref{thm:main}(i). 
\\
\par We now turn to part (ii) of the theorem and divide the proof into three 
steps. 
\\
\\
{\sc Step 1:} There exists a $(2, \, 2)$-almost AP-destroying permutation of 
$\mathbb{Z}_r$ for some $r$. 
\\
\\
\emph{Proof.} Let $\pi_1,\,\dots,\,\pi_6$ be as above. Using 
Table \ref{tab:data} and Lemma \ref{lem:normalise}, it is easy to check that 
we can also find permutations $\pi_7,\,\dots,\,\pi_{15}$ of 
$\mathbb{Z}_{n_{7}},\,\dots,\,\mathbb{Z}_{n_{15}}$ respectively which collectively
destroy all of the patterns 
$s \mapsto t$, $s \in \{\pm 2\}$, $t \in \{0, \, \pm 1, \, \pm 2\}$. Hence, by 
a similar argument to above, there exists a $(2, \, 2)$-almost AP-destroying 
permutation $\chi_{r}$ of $\mathbb{Z}_r$, where $r = \prod_{i=1}^{15} n_i$.
\\
\\
{\sc Step 2:} For every $s, \, t \in \mathbb{N}_0$ there exists 
$r_{0}(s,\,t) \in \mathbb{N}$ and an $(s, \, t)$-almost AP-destroying 
permutation of $\mathbb{Z}_{r_{0}(s, \, t)}$. 
\\
\\ 
\emph{Proof.} Let $m$ be an integer such that there exists an AP-destroying 
permutation of $\mathbb{Z}_m$, and let $\chi_{m}$ be such a permutation. 
Identify $\mathbb{Z}_{N}$ with the set $\{0,\,1,\,\dots,\,N-1\}$ for any $N$. 
Let $r = r_{0}(2, \, 2)$ be as in Step 1. Then (see Lemma 3.5 of \cite{H}) the 
map $\chi_{rm}: \mathbb{Z}_{rm} \rightarrow \mathbb{Z}_{rm}$ given by 
\begin{equation*}\label{eq:rm22}
\chi_{rm}(rx+y) = r \chi_{m}(x) + \chi_{r}(y), \;\; 0 \leq x < m, \;\; 
0 \leq y < r,
\end{equation*}
is easily seen to be $(2,\,2)$-almost AP-destroying, provided $m > 2$. Thus, 
by the already proven Theorem \ref{thm:main}(i), there exists a 
$(2, \, 2)$-almost AP-destroying permutation $\chi_{rm}$ of $\mathbb{Z}_{rm}$, 
for all sufficiently large $m$. Moreover, after a suitable translation, we can 
choose $\chi_{rm}$ so that $rm - 1$ is a fixed point. In this case, the 
restriction of $\chi_{rm}$ to $\{0,\,1,\,\dots,\,rm-2\}$ can be considered as 
a permutation of $\mathbb{Z}_{rm-1}$ and, since $\chi_{rm}$ was 
$(2, \, 2)$-almost AP-destroying, it is easily seen that this restriction is 
$(1, \, 1)$-almost AP-destroying. To summarise, we have shown that there is an 
infinite arithmetic progression $\mathcal{A}$, consisting of numbers congruent 
to $-1 \; ({\hbox{mod $r$}})$, such that there exists a $(1, \, 1)$-almost 
AP-destroying permutation of $\mathbb{Z}_n$ for all $n \in \mathcal{A}$. 
Furthermore, since the first term and common difference of $\mathcal{A}$ are 
relatively prime, there is an infinite subsequence $a_1,\,a_2,\dots$ of 
elements of $\mathcal{A}$ consisting of pairwise relatively prime numbers. 
This follows from Dirichlet's theorem, though it is actually trivial to prove, 
in a similar manner to Euclid's proof of the existence of infinitely many 
primes.
\par Now fix $s, \, t \in \mathbb{N}_0$. For a permutation of some 
$\mathbb{Z}_n$ to be $(s, \, t)$-almost AP-destroying, it just needs to 
destroy a finite number of patterns. Using Lemma \ref{lem:normalise} it 
follows that there exists an $i = i(s,\,t)$ and $(1,\,1)$-almost AP-destroying 
permutations $\chi_{j}$ of $\mathbb{Z}_{a_j}$, $j = 1,\,\dots,\,i$, which 
collectively destroy every pattern $s^{\prime} \mapsto t^{\prime}$, 
$|s^{\prime}| \leq s$, $|t^{\prime}| \leq t$. Then, by a construction similar to 
(\ref{eq:master}), we can construct an $(s,\,t)$-almost AP-destroying 
permutation of $\mathbb{Z}_{r_{0}(s,\,t)}$, where 
$r_{0}(s,\,t)= \prod_{j=1}^{i(s,\,t)} a_j$.
\\
\\
{\sc Step 3:} Theorem \ref{thm:main}(ii) holds.
\\
\\
\emph{Proof.} Clearly, it suffices to prove the theorem when $s=t$ and $t=0$ 
has already been dealt with. So fix $t > 0$. We claim the theorem holds with 
\begin{equation*}\label{eq:nzerost}
n_{0}(s, \, t) = \left[ r_{0}(4t+7, \, 4t+7) \right]^2.
\end{equation*}
To simplify notation, set $M := r_{0}(4t + 7, \, 4t+7)$ and fix $n \geq M^2$. 
Our task is to construct a $(t, \, t)$-almost AP-destroying permutation of 
$\mathbb{Z}_n$. Write $n = kM + l$, $0 \leq l < M$ and place the numbers 
$0,\,1,\,\dots,\,n-1$ clockwise around a circle. As before, we find it most 
convenient to describe our permutation in terms of a rearrangement of this 
circular string of $n$ numbers. The rearrangement will be broken down into 4 
stages, of which stages 2 and 3 correspond to the procedure in the proof of 
part (i) of Theorem \ref{thm:main}, while stages 1 and 4 deal with the fact 
that $t > 0$. Stage 4 is essentially the ``reverse'' of Stage 1. 
\\
\\
\emph{Stage 1:} First divide the $n$ numbers into $M$ consecutive clockwise 
blocks $B_0,\,\dots,\,B_{M-1}$, each of size $\lfloor n/M \rfloor$ or 
$\lceil n/M \rceil$. Unlike in the proof of part (i), here it doesn't matter 
which blocks have which size, as we will only appeal in the end to Claim 1 
from earlier. Let $\beta_{1}(x)$ denote the number of the block containing 
$x \in [0,\,n)$ at this point. 
\par Next, partition the blocks $B_i$ into 
$\lfloor \frac{M}{t+1} \rfloor$ ``superblocks'' 
$C_0,\,\dots,\,C_{\lfloor M/(t+1) \rfloor}$, each consisting of either $t+1$ or 
$t+2$ ordinary blocks, with the larger superblocks placed furthest clockwise 
from zero. Thus 
\begin{equation}\label{eq:superblocks}
C_0 = (B_0,\,\dots,\,B_{t}), \;\;\; C_{1} = (B_{t+1},\,\dots,\,B_{2t+1}), 
\;\;\; {\hbox{etc.}}
\end{equation}
Note that since $M$ is extremely large compared to $t$, there is no problem in 
making this subdivision of ordinary blocks.
Now rearrange the individual numbers in each superblock in such a way that, if 
the superblock contains $t+i$ ordinary blocks, $i \in \{1,\,2\}$, then after 
this rearrangement, the numbers inside any ordinary block will form an AP of 
common difference $t+i$. For example, consider the superblock $C_0$. There is 
a unique reordering $(i_0,\,i_1,\,\dots,\,i_t)$ of $(0,\,1,\,\dots,\,t)$ such 
that$|B_{i_0}| \geq |B_{i_1}| \geq \dots \geq |B_{i_t}|$ and the indices are 
increasing as long as the block sizes are constant. After rearrangement, 
$B_{i_0}$ would contain $0,\,t+1,\,2(t+1),\dots$, $B_{i_1}$ would contain 
$1,\,t+2,\,2t+3,\dots$ and so on up to $B_{i_t}$ which would contain 
$t,\,2t+1,\dots$. 
\par Let $\beta_{2}(x)$ denote the (ordinary) block containing $x$ at this 
point and note that
\begin{equation}\label{eq:betadiff}
\left| \beta_{2}(x) - \beta_{1}(x) \right| \leq t+1,
\end{equation}
where plus one comes from the fact that some superblocks may contain $t+2$ 
ordinary blocks. 
\\
\\
\emph{Stage 2:} Choose a $(4t+7, \, 4t+7)$-almost AP-destroying permutation 
$\pi_0$ of $\mathbb{Z}_M$ and permute the ordinary blocks according to 
this - in other words, after applying $\pi_0$ the blocks 
$B_{\pi_{0}^{-1}(0)},\,\dots,\,B_{\pi_{0}^{-1}(M-1)}$ appear in clockwise order. Let 
$\mathcal{B}_i := B_{\pi_{0}^{-1}(i)}$ and let $\beta_{3} (x)$ denote the scripted 
block containing $x$ at this point. Thus 
\begin{equation}\label{eq:beta3}
\beta_{3} (x) = \pi_{0} (\beta_{2}(x)). 
\end{equation}
\emph{Stage 3:} Let $\tau^{0}, \, \tau^{1}$ be AP-destroying permutations
of $\left\{1,\,\dots,\, \lfloor \frac{n}{M} \rfloor \right\}$ and 
$\left\{1,\,\dots,\,\lceil \frac{n}{M} \rceil \right\}$ respectively, 
considered as subsets of $\mathbb{N}$. From Proposition 2.3(ii) of \cite{H} 
we know that such permutations exist. Given any set $S$ of integers which 
forms an AP of length $\lfloor n/M \rfloor$ (resp. $\lceil n/M \rceil$), it is 
obvious how to extract from $\tau^{0}$ (resp. $\tau^{1}$) an AP-destroying 
permutation of $S$. We perform such a permutation on each scripted block 
$\mathcal{B}_i$. 
\\
\\
\emph{Stage 4:} Divide the scripted blocks into superblocks in the same way as 
in (\ref{eq:superblocks}), thus
\begin{equation*}
\mathcal{C}_0 = (\mathcal{B}_0,\,\dots,\,\mathcal{B}_{t}), \;\;\; 
\mathcal{C}_{1} = (\mathcal{B}_{t+1},\,\dots,\,\mathcal{B}_{2t+1}), \;\;\; 
{\hbox{etc.}}
\end{equation*}
We then rearrange the numbers in each superblock in such a way that, for each 
block $\mathcal{B}_i$, the positions of its elements after rearrangement form 
an AP of common difference $|\mathcal{C}_0| \in \{t+1, \, t+2\}$. This is 
accomplished by reversing the procedure in Stage 1 - we hope it is clear what 
is meant by this and spare the reader further details. Let $\beta_{4}(x)$ be 
the number of the scripted block containing $x$ at this point and note that, 
analogous to (\ref{eq:betadiff}), one has
\begin{equation}\label{eq:scriptbetadiff}
\left| \beta_{4}(x) - \beta_{3}(x) \right| \leq t+1.
\end{equation}
\par Let $\pi: \mathbb{Z}_n \rightarrow \mathbb{Z}_n$ be the permutation 
defined by the rearrangement accomplished in Stages 1-4, that is, $\pi(x)$ 
denotes the location of the number $x$ in the string after the rearrangement. 
We claim that $\pi$ is $(t,\,t)$-almost AP-destroying. To see this, let 
$(a,\,b,\,c)$ be a triple of elements of $\mathbb{Z}_n$ satisfying
\par (I) $a, \, b, \, c$ not all equal,
\par (II) $a + c -2b \equiv \eta \; ({\hbox{mod $n$}})$ for some 
$\eta \in [-t, \, t]$
\\
and consider two cases:
\\
\\
{\sc Case 1:} $a, \, b, \, c$ all lie in the same ordinary block $B_i$ at the 
end of Stage 1. 
\\
\\
Since $a, \, b, \, c$ are not all equal and the numbers in $B_i$, as it stands 
after Stage 1, form an AP with common difference strictly greater than $t$, 
property (II) can only hold if $a, \, b, \, c$ form a non-trivial AP modulo 
$n$. But since the appropriate $\tau^{j}$ destroys APs of integers, and hence 
also APs modulo $n$ since $n$ is much larger than $M$, we see that the 
locations of $a, \, b, \, c$ will not form an AP modulo $n$ after Stage 3. But 
they will still lie in the same scripted block hence, after Stage 4, 
$\pi(a)+\pi(c)-2\pi(b)$ must be a non-zero multiple of 
$t+i$, $i \in \{1,\,2\}$. In particular, 
$\pi(a) + \pi(c) -2\pi(b) \; ({\hbox{mod $n$}}) \not\in [-t, \, t]$. 
\\
\\
{\sc Case 2:} $a, \, b, \, c$ are not all in the same ordinary block upon 
completion of Stage 1.
\\
\\
By definition, what we're assuming in this case is that 
$\beta_{2}(a), \, \beta_{2}(b), \, \beta_{2}(c)$ are not all equal. If 
$a, \, b, \, c$ formed an AP modulo $n$ then it would follow from Claim 1 in 
the proof of part (i) of Theorem \ref{thm:main} that 
\begin{equation*}\label{eq:firstest}
\beta_{1}(a) + \beta_{1}(c) - 2\beta_{1}(b) \; ({\hbox{mod $M$}}) \in 
[-2, \, 2].
\end{equation*}
Given that (II) holds and that each ordinary block has size greater than $t$, 
we can at least be sure that
\begin{equation*}\label{eq:secondest}
\beta_{1}(a) + \beta_{1}(c) - 2 \beta_{1}(b) \; ({\hbox{mod $M$}}) \in 
[-3, \, 3].
\end{equation*}
Combined with (\ref{eq:betadiff}) it follows that
\begin{equation*}\label{eq:thirdest}
\beta_{2}(a) + \beta_{2}(c) - 2\beta_{2}(b) \; ({\hbox{mod $M$}}) \in 
[-(4t+7), \, 4t+7].
\end{equation*}
But the permutation $\pi_0$ is $(4t+7, \, 4t+7)$-almost AP-destroying and 
thus, by (\ref{eq:beta3}), 
\begin{equation*}\label{eq:afterest}
\beta_{3}(a) + \beta_{3}(c) - 2\beta_{3}(b) \; ({\hbox{mod $M$}}) \not\in 
[-(4t+7), \, 4t+7].
\end{equation*}
By (\ref{eq:scriptbetadiff}), this implies in turn that
\begin{equation*}\label{eq:diffover3}
\beta_{4}(a) + \beta_{4}(c) - 2 \beta_{4}(b) \; ({\hbox{mod $M$}}) \not\in 
[-3, \, 3].
\end{equation*}
Since the scripted blocks still have size greater than $t$, it follows from 
Claim 1 on page 4 that 
$\pi(a) + \pi(c) - 2\pi(b) \; ({\hbox{mod $n$}}) \not\in [-t, \, t]$, as 
desired. 
\qed

\section{Proof of Theorem \ref{thm:primes}}\label{sect:pfprimes}

Let $p$ be a prime. We denote by $\mathcal{R}_p$ (resp. $\mathcal{N}_p$)
the collection of quadratic residues (resp. non-residues) modulo $p$. We will
be slightly abusive in this context and use the same notations to denote 
subsets of $\mathbb{Z}_p$ and of $\mathbb{Z}$. Hence, as subsets of 
$\mathbb{Z}_p$ one has
\begin{equation*}\label{eq:residues}
\mathcal{R}_p = \{x^2 : x \in \mathbb{Z}_p \}, \;\;\;\; \mathcal{N}_p = 
\mathbb{Z}_p \backslash \mathcal{R}_p,
\end{equation*}
whereas, as subsets of $\mathbb{Z}$,
\begin{equation*}\label{eq:intresidues}
\mathcal{R}_p = \left\{x \in \mathbb{Z} : \left( \frac{x}{p} \right) \in 
\{0,\,1\} \right\}, \;\;\;\; \mathcal{N}_p = 
\mathbb{Z} \backslash \mathcal{R}_p = \left\{x \in \mathbb{Z} : \left( 
\frac{x}{p} \right) = -1 \right\}.
\end{equation*}

\begin{lemma}\label{lem:othree}
Let $p$ be a prime such that $p \equiv 3 \; ({\hbox{mod $8$}})$. Then 
both $-1$ and $2$ are in $\mathcal{N}_p$.
\end{lemma}

\begin{proof}
This is elementary number theory. That $-1$ is not a square mod $p$ follows
from Lagrange's theorem for groups. That $2$ is not a square follows from
Gauss' Lemma.
\end{proof}

\begin{lemma}\label{lem:ofour}
Let $p > 3$ be a prime. Then there exists an integer $\xi$ such that both
$\xi$ and $\xi - 1$ lie in $\mathcal{N}_p$.
\end{lemma}

\begin{proof}
This follows immediately from the fact that 
$|\mathcal{N}_p| = |\mathcal{R}_p| - 1$ and 
$\{0,\,1,\,4\} \subseteq \mathcal{R}_p$.
\end{proof}

\begin{lemma}\label{lem:ofive}
Let $p > 2$ be a prime, and let $a,\,b,\,c$ be integers not divisible by $p$. 
Then $(0,\,0)$ is the only solution in $\mathbb{Z}_p$ to the 
congruence $ax^2 + bxy + cy^2 \equiv 0 \; ({\hbox{mod $p$}})$ if and only
if $b^2 - 4ac \in \mathcal{N}_p$.
\end{lemma}

\begin{proof}
Elementary. Suppose we have a solution with $y \not\equiv 0$. Then 
completion of squares gives 
\begin{equation*}\label{eq:disc}
xy^{-1} \equiv (2a)^{-1} \left( -b \pm \sqrt{b^2 - 4ac} \right),
\end{equation*}
which is meaningful if and only if $b^2 - 4ac \in \mathcal{R}_p$. 
\end{proof}

We are now ready to prove Theorem \ref{thm:primes}. Let $p > 3$ be a prime 
congruent to 
$3 \; ({\hbox{mod $8$}})$ and let $\xi$ be any integer satisfying the
conditions of Lemma \ref{lem:ofour}.
Define the map $f: \mathbb{Z}_p \rightarrow \mathbb{Z}_p$ as follows:
\begin{equation*}\label{eq:f}
f(x) = \left\{ \begin{array}{lr} x^2, & 
{\hbox{if $x \in \{0,\,2,\, 4,\,\dots,\,p-1\}$}}, 
\\ \xi x^2, & {\hbox{if $x \in \{1,\,3,\,5,\, \dots, \,p-2\}$}}. 
\end{array} \right.
\end{equation*}
Note that $f$ is one-to-one. Let $(a,\, b,\, c)$ be an arithmetic 
progression modulo $p$. Denote $a := x$, $b := x+y$, $c := x+2y$.
We need to show that
\begin{equation}\label{eq:apf}
f(x) + f(x+2y)\equiv 2f(x+y) \Rightarrow y\equiv 0.
\end{equation}
Denote also $\mathbb{E} := \{0,\, 2,\, 4,\, \dots, \, p-1\}$ and 
$\mathbb{O} := \{1,\, 3,\, 5,\, \dots, \, p-2\}$. We 
do a case-by-case analysis. 
\\
\\
{\sc Case 1:} $a,\, b,\, c \in \mathbb{E}$. 
\\
\\
The congruence in (\ref{eq:apf}) becomes 
\begin{equation*}\label{eq:cong}
x^2 + (x+2y)^2 \equiv 2 (x+y)^2,
\end{equation*}
which reduces to $2y^2 \equiv 0$, hence $y \equiv 0$ since $p > 2$. Thus
(\ref{eq:apf}) holds in this case.
\\
\\
The case when $a,\, b,\, c \in \mathbb{O}$ is completely analogous to Case 1.
\\
\\
{\sc Case 2:} $a,\, b \in \mathbb{E}$, $c \in \mathbb{O}$.
\\
\\
The congruence in (\ref{eq:apf}) becomes
\begin{equation*}\label{eq:cong2}
x^2 + \xi (x+2y)^2 \equiv 2(x+y)^2,
\end{equation*}
which can be expanded as
\begin{equation}\label{eq:exp2}
(\xi -1) x^2 + 4(\xi - 1)xy + 2(2\xi -1) y^2 \equiv 0.
\end{equation}
By the choice of $\xi$ we know that $\xi-1 \in \mathcal{N}_p$, so
in particular $\xi -1 \not\equiv 0$. Also, $2\xi -1 \not\equiv 0$, 
for otherwise we would have $\xi - 1 \equiv -1/2$, contradicting 
Lemma \ref{lem:othree} and the fact that $\xi - 1 \in \mathcal{N}_p$. 
Hence all the 
coefficients in the binary quadratic form in (\ref{eq:exp2}) 
are non-zero modulo $p$,
so we can apply Lemma \ref{lem:ofive} and deduce that there is no solution with
$y \not\equiv 0$ if and only if 
$[4(\xi-1)]^2 - 8(\xi-1)(2\xi-1) \in \mathcal{R}_p$. But
\begin{equation*}\label{eq:shit2}
[4(\xi - 1)]^2 - 8(\xi-1)(2\xi -1) = [4(\xi-1)]^2 
\cdot \left( \frac{-1}{2(\xi -1)} \right),
\end{equation*}
hence this lies in $\mathcal{R}_p$ if and only if $\frac{-1}{2(\xi-1)}$
does so. But the latter contradicts Lemma \ref{lem:othree} 
and the choice of $\xi$. 
\\
\\
The case when $a \in \mathbb{O}$, $b,\, c \in \mathbb{E}$ is
completely analogous to Case 2. 
\\
\\
{\sc Case 3:} $a,\, c \in \mathbb{E}$, $b \in \mathbb{O}$.
\\
\\
The congruence in (\ref{eq:apf}) becomes
\begin{equation*}\label{eq:cong3}
x^2 + (x+2y)^2 \equiv 2\xi (x+y)^2,
\end{equation*}
which can be expanded as
\begin{equation*}\label{eq:exp3}
(\xi -1) x^2 + 2(\xi - 1)xy + (\xi -2) y^2 \equiv 0.
\end{equation*}
Once again, all the coefficients are non-zero modulo $p$,
so we can apply Lemma \ref{lem:ofive} and deduce that there is no solution with
$y \not\equiv 0$ if and only if $[2(\xi-1)]^2 - 4(\xi-1)(\xi-2) \in
\mathcal{R}_p$. But
\begin{equation*}\label{eq:shit3}
[2(\xi - 1)]^2 - 4(\xi-1)(\xi -2) = [2(\xi-1)]^2 
\cdot \left( \frac{1}{\xi -1} \right),
\end{equation*}
hence this lies in $\mathcal{R}_p$ if and only if $\frac{1}{\xi-1}$
does so. But the latter contradicts the choice of $\xi$. 
\\
\\
{\sc Case 4:} $a,\, b \in \mathbb{O}$, $c \in \mathbb{E}$.
\\
\\
The congruence in (\ref{eq:apf}) becomes
\begin{equation*}\label{eq:cong4}
\xi x^2 + (x+2y)^2 \equiv 2\xi (x+y)^2,
\end{equation*}
which can be expanded as
\begin{equation*}\label{eq:exp4}
(\xi -1) x^2 + 4(\xi - 1)xy + 2(\xi -2) y^2 \equiv 0.
\end{equation*}
The coefficients are all non-zero modulo $p$ so,
by Lemma \ref{lem:ofive}, there is no solution with
$y \not\equiv 0$ if and only if $[4(\xi-1)]^2 - 8(\xi-1)(\xi-2) \in
\mathcal{R}_p$. But
\begin{equation*}\label{eq:shit4}
[4(\xi - 1)]^2 - 8(\xi-1)(\xi -2) = [4(\xi-1)]^2 
\cdot \left( \frac{\xi}{2(\xi -1)} \right),
\end{equation*}
hence this lies in $\mathcal{R}_p$ if and only if $\frac{\xi}{2(\xi-1)}$
does so. Once again, this contradicts Lemma \ref{lem:othree} 
and the choice of $\xi$. 
\\
\\
The case when $a \in \mathbb{E}$, $b,\, c \in \mathbb{O}$ is
completely analogous to Case 4. 
\\
\\
{\sc Case 5:} $a,\, c \in \mathbb{O}$, $b \in \mathbb{E}$.
\\
\\
The congruence in (\ref{eq:apf}) becomes
\begin{equation*}\label{eq:cong5}
\xi x^2 + \xi (x+2y)^2 \equiv 2(x+y)^2,
\end{equation*}
which can be expanded as
\begin{equation*}\label{eq:exp5}
(\xi -1) x^2 + 2(\xi - 1)xy + (2\xi -1) y^2 \equiv 0.
\end{equation*}
The coefficients are still non-zero modulo $p$ so,
by Lemma \ref{lem:ofive}, there is no solution with
$y \not\equiv 0$ if and only if $[2(\xi-1)]^2 - 4(\xi-1)(2\xi-1) \in
\mathcal{R}_p$. But
\begin{equation*}\label{eq:shit5}
[2(\xi - 1)]^2 - 4(\xi-1)(2\xi -1) = [2(\xi-1)]^2 
\cdot \left( \frac{-\xi}{\xi -1} \right),
\end{equation*}
hence this lies in $\mathcal{R}_p$ if and only if $\frac{-\xi}{\xi-1}$
does so. Once again we have a contradiction to 
Lemma \ref{lem:othree} and the choice of $\xi$. This covers all possible cases 
and completes the proof of Theorem \ref{thm:primes}.
\qed

\setcounter{equation}{0}

\section{Final remarks}\label{sect:final}

We don't know of any reason to suspect that Conjecture C of \cite{H} may be 
false. However, a full proof of it remains a challenging problem, given that 
the value of $n_0$ in Theorem \ref{thm:main}(i) still leaves a brute-force 
computational attack way out of reach. A less ambitious goal could be to see 
how small $n_0$ can be made using the ideas of this paper.
\par An obvious approach to Conjecture C is to consider a uniformly random 
permutation $\pi$ of $\mathbb{Z}_n$ and let $X$ be the number of non-trivial 
APs not destroyed by $\pi$. It is easy to check that 
$\mathbb{E}[X] =  \Theta (n)$. As a first hypothesis, one might guess that the 
events that individual APs are not destroyed by $\pi$ are almost independent 
and hence that $X$ is approximately Poisson distributed. In that case, the 
proportion of AP-destroying permutations of $\mathbb{Z}_n$ would decrease 
exponentially with $n$. A priori, this approach could still only yield 
Conjecture C for $n$ sufficiently large, but probably with a value of $n_0$ 
which is sufficiently small to complete the proof by direct computation. 
However, it remains an open question whether this intuition can be made 
rigorous. A successful moment analysis \emph{was} carried out in \cite{JS} for 
permutations destroying APs of length $4$, with the important point being that 
the expected number of non-trivial $4$-term APs not destroyed by a random 
permutation of $\mathbb{Z}_n$ is $O(1)$. 
This provides some additional circumstantial evidence for Conjecture C. 
\par An AP of length $4$ is a common solution to the pair of linear equations 
$x_1 - 2x_2 + x_3 = 0$, $x_2 - 2x_3 + x_4 = 0$. It is natural to ask the 
following more general question:

\begin{question}\label{quest:genequation}
Let $k,\,m \in \mathbb{N}$ and let 
$\mathcal{L}_{i}(x_1,\,\dots,\,x_k) = 0$, $i = 1,\,\dots,\,m$, be linear 
equations with integer coefficients. When is it the case that there is an 
$n_0 = n_0 (\mathcal{L}_1,\,\dots,\,\mathcal{L}_m)$ such that there exists a 
permutation of $\mathbb{Z}_n$ destroying all non-trivial solutions 
(as defined in \cite{R}) to $\mathcal{L}_1 = \dots = \mathcal{L}_m = 0$ for 
all $n \geq n_0$ ?
\end{question}

Indeed, it seems reasonable even to ask this kind of question for general 
polynomial equations, not just linear ones. Here we confine further 
speculation to the case of a single linear equation 
$a_0 + \sum_{i=1}^{k} a_i x_i = 0$, $a_i \neq 0 \; \forall \, i > 0$. Recall 
that the equation is said to be \emph{(translation) invariant} if 
$a_0 = \sum_{i=1}^{k} a_i = 0$. One can make the following straightforward 
observations:
\\
\par (i) Suppose $a_0 \neq 0$. If $n > 2|a_0|$, then there exists a unit 
$u \in \mathbb{Z}_{n}^{\times}$ such that 
$ua_0 \not\equiv a_0 \; ({\hbox{mod $n$}})$. Then the permutation 
$x \mapsto ux \; ({\hbox{mod $n$}})$ will destroy all solutions to the 
equation. 
\par (ii) Suppose $\sum_{i=1}^{k} a_i \neq 0$. If 
$n > \left| \sum_{i=1}^{k} a_i \right|$ then the translation 
$x \mapsto x+1 \; ({\hbox{mod $n$}})$ will destroy all solutions to the 
equation.
\par (iii) Suppose the equation is invariant and $k = 2$, so the equation reads
$a_1 (x_1 - x_2) = 0$, for some $a_1 \neq 0$. A permutation $\pi$ of 
$\mathbb{Z}_n$ will destroy all non-trivial solutions if and only if, 
whenever $x_1$ and $x_2$ are distinct but lie in the same congruence class 
modulo $\frac{n}{{\hbox{GCD}}(a_1,\,n)}$, then $\pi(x_1)$ and $\pi(x_2)$ lie 
in different congruence classes. Clearly, such a permutation exists for all 
$n \geq \left[{\hbox{GCD}}(a_1, \, n)\right]^2$. 
\\
\par
Hence, for a single linear equation, Question \ref{quest:genequation} is only 
interesting if the equation is invariant and $k \geq 3$. If we then consider a 
uniformly random permutation of $\mathbb{Z}_n$, it is easy to see that the 
expected number of non-trivial solutions not destroyed is $\Theta (n^{k-2})$. 
This suggests that permutations destroying all non-trivial solutions should 
exist when $k = 3$, but perhaps do not do so at all when $k > 3$. We therefore 
ask the following:

\begin{question}\label{quest:singleeq}
Let $\mathcal{L}(x_1,\,\dots,\,x_k) = a_0 + \sum_{i=1}^{k} a_i x_i = 0$, 
$a_i, \in \mathbb{Z}$, $a_i \neq 0 \; \forall \, i > 0$, be a linear equation. 
Is it true that the following statements are equivalent: 
\par (i) There is an $n_0 = n_{0}(\mathcal{L})$ such that, for every 
$n \geq n_0$, there exists a permutation $\pi$ of $\mathbb{Z}_n$ destroying 
all non-trivial solutions of $\mathcal{L} = 0$. 
\par (ii) Either the equation $\mathcal{L} = 0$ is variant, or it is invariant 
and $k \in \{2, \, 3\}$ ?
\end{question} 

As a final remark, note that in Proposition 2.3(i) of \cite{H} we proved that 
no permutation of any finite abelian group can destroy all non-trivial 
solutions to the Sidon equation $x_1 + x_2 - x_3 - x_4 = 0$. However, we do 
not see at this point how to modify that argument for equations in four or 
more variables in general. 




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\end{document}
