On Sequence Groups

TL;DR

This paper investigates the algebraic structure of second order recursive sequences, introducing the sequence group, analyzing prime divisors, and simplifying algebraic properties with rational parameters, leading to new insights on divisibility and density of prime divisors.

Contribution

It introduces the sequence group with simplified algebraic structure, studies prime divisors via the group modulo p, and establishes conditions for divisibility, advancing understanding of recursive sequence properties.

Findings

Sequence group is cyclic of order p±1 mod p.

Prime divisors relate to quadratic residues of the sequence norm.

Numerical experiments suggest prime divisor density around 0.35.

Abstract

Linear second order recursive sequences with arbitrary initial conditions are studied. For sequences with the same parameters a ring and a group is attached, and isomorphisms and homomorphisms are established for related parameters. In the group, called the {\it sequence group}, sequences are identified if they differ by a scalar factor, but not if they differ by a shift, which is the case for the Laxton group. Prime divisors of sequences are studied with the help of the sequence group $\mod p$, which is always cyclic of order $p\pm 1$. Even and odd numbered subsequences are given independent status through the introduction of one rational parameter in place of two integer parameters. This step brings significant simplifications in the algebra. All elements of finite order in Laxton groups and sequence groups are described effectively. A necessary condition is established for a prime $p$ to be a divisor of a sequence: {\it the norm (determinant) of the respective element of the ring must be a quadratic residue $\mod p$}. This leads to an uppers estimate of the set of divisors by a set of prime density $1/2$. Numerical experiments show that the actual density is typically close to $0.35$. A conjecture is formulated that the sets of prime divisors of the even and odd numbered elements are independent for a large family of parameters.