Mod $p$ equivalence classes of linear recurrence sequences of degree~$2$

Abstract

Laxton introduced a group structure on the set of equivalence classes of linear recurrence sequences of degree~2. This result yields much information on the divisibilities of such sequences. In this paper, we introduce other equivalence relations for the set of linear recurrence sequences $(G_n)$, which are defined by $G_0, G_1 \in \mathbb{Z} $ and $G_n=TG_{n-1}-NG_{n-2}$ for fixed integers~$T$ and $N=\pm 1$. The relations are given by certain congruences modulo~$p$ for a fixed prime number~$p$, which are different from Laxton's without modulo $p$ equivalence relations. We determine the initial terms $G_0$ and $G_1$ of all of the representatives of the equivalence classes $\overline {(G_n)}$ satisfying $p\nmid G_n$ for any integer~$n$ and give the number of equivalence classes. Furthermore, we determine the representatives of Laxton's without modulo~$p$ classes from our modulo~$p$ classes.