Some Narcissistic Power-Sequence \(\mathbb{Z}_{n+1}\) Terraces with \(n\) an Odd Prime Power

Abstract

A terrace for \(\mathbb{Z}_m\) is an arrangement \((a_1, a_2, \ldots, a_m)\) of the \(m\) elements of \(\mathbb{Z}_m\) such that the sets of differences \(a_{i+1} – a_i\) and \(a_i – a_{i+1}\) (\(i = 1, 2, \ldots, m-1\)) between them contain each element of \(\mathbb{Z}_m \setminus \{0\}\) exactly twice. For \(m\) odd, many procedures are available for constructing power-sequence terraces for \(\mathbb{Z}_m\); each such terrace may be partitioned into segments one of which contains merely the zero element of \(\mathbb{Z}_m\) whereas each other segment is either (a) a sequence of successive powers of a non-zero element of \(\mathbb{Z}_m\) or (b) such a sequence multiplied throughout by a constant. For \(n\) an odd prime power satisfying \(n \equiv 1\) or \(3 \pmod{8}\), this idea has previously been extended by using power-sequences in \(\mathbb{Z}_n\) to produce some \(\mathbb{Z}_m\) terraces \((a_1, a_2, \ldots, a_m)\) where \(m = n+1 = 2^\mu\), with \(a_{i+1} – a_i = -(a_{i+1+\mu} – a_{i+\mu})\) for all \(i \in [1, \mu-1]\). Each of these “da capo directed terraces” consists of a sequence of segments, one containing just the element \(0\) and another just containing the element \(n\), the remaining segments each being of type (a) or (b) above with each of its distinct entries \(z\) from \(\mathbb{Z}_n \setminus \{0\}\) evaluated so that \(1 \leq x \leq n-1\). Now, for many odd prime powers \(n\) satisfying \(n \equiv 1 \pmod{4}\), we similarly produce narcissistic terraces for \(\mathbb{Z}_{n+1}\); these have \(a_{i+1} – a_i = a_{m-i+1} – a_{m-i}\) for all \(i \in [1, \mu-1]\).