Published online by Cambridge University Press: 03 July 2019
A Ducci sequence is a sequence of integer $n$-tuples generated by iterating the map
$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$
$P(n)$ the maximal period of such sequences for given
$n$. We prove a new upper bound in the case where
$n$ is a power of a prime
$p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$ for which
$2$ is a primitive root and the Pellian equation
$x^{2}-py^{2}=-4$ has no solutions in odd integers
$x$ and
$y$.