Complete Padovan sequences in finite fields

Abstract

Given a prime $p\ge 5$, and given $1<\kappa<p-1$, we call a sequence $(a_n)_{n}$ in $\mathbb{F}_p$ a $\Phi_{\kappa}$-sequence if it is periodic with period $p-1$, and if it satisfies the linear recurrence $a_n+a_{n+1}=a_{n+\kappa}$ with $a_0=1$. Such a sequence is said to be a complete $\Phi_{\kappa}$-sequence if in addition $\{a_0,a_1,...,a_{p-2}\}=\{1,...,p-1\}$. For instance, every primitive root $b$ mod $p$ generates a complete $\Phi_{\kappa}$-sequence $a_n=b^n$ for some (unique) $\kappa$. A natural question is whether every complete $\Phi_{\kappa}$-sequence is necessarily defined by a primitive root. For $\kappa=2$ the answer is known to be positive. In this paper we reexamine that case and investigate the case $\kappa=3$ together with the associated cases $\kappa=p-2$ and $\kappa=p-3$.