Further results on logarithmic terraces

Abstract

Let p be an odd prime, and let x be a primitive root of p. Suppose that we write the elements of ${\mathbb{Z}}_{p-1}$ as $1,2,\dots ,p-1,$ and that, wherever we evaluate $x^l(\mathrm{mod}p)$, we always write it as one of $1,2,\dots ,p-1.$ Let $\ell =(l_1,\dots ,l_{p-1})$ be a terrace for ${\mathbb{Z}}_{p-1}$. Then $\ell$ is said to be a logarithmic terrace if $\mathbf{e}=(e_1,\dots ,e_{p-1})$, defined by $e_i\equiv x^{l_i}(\mathrm{mod}p)$, is also a terrace for ${\mathbb{Z}}_{p-1}$. We study properties of logarithmic terraces, in particular investigating terraces which are simultaneously logarithmic for two different primitive roots.