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A Ducci sequence is a sequence of integer $n$-tuples obtained 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 odd

$n$. We prove a lower bound for

$P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo

$n$.

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

Using a recent result on the sum–product problem, we estimate the number of elements γ in a prime finite field such that both γ and γ+γ−1 are of small order.

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LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES

ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS

SUM–PRODUCT ESTIMATES AND MULTIPLICATIVE ORDERS OF γ AND γ+γ−1 IN FINITE FIELDS

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3 results

LOWER BOUNDS FOR PERIODS OF DUCCI SEQUENCES

ELEMENTS OF LARGE ORDER IN PRIME FINITE FIELDS

SUM–PRODUCT ESTIMATES AND MULTIPLICATIVE ORDERS OF γ AND γ+γ−1 IN FINITE FIELDS