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A Ducci sequence is a sequence of integer $n$-tuples generated by iterating the map

$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$

Such a sequence is eventually periodic and we denote by

$P(n)$ 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

$y$.

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