On Sequence Groups

TL;DR

This work develops a ring-and-group-theoretic framework for linear second-order recurrences, introducing the rings $\mathcal{R}(T,Q)$ and the one-parameter form $\mathcal{R}(t)$, together with the sequence group $\mathcal{L}(t)$ and the Laxton group $\mathcal{G}(t)$. It shows that the mod $p$ sequence groups $\mathcal{L}_p(t)$ are cyclic of order $p\pm1$, and establishes a necessary condition for a prime to divide a sequence: the determinant must be a quadratic residue mod $p$. The paper analyzes even/odd subsequences via the parameter $t$, studies the torsion structure of Laxton groups across circular, cubic, and non-cyclotomic cases, and develops a detailed prime-divisor framework using the sets $\Gamma_X$ and their densities. A central and novel contribution is the Independence Conjecture, predicting near-independence of prime divisors between even and odd subsequences for a broad class of primitive parameters, supported by substantial theoretical bounds and numerical evidence. The results provide structural insight into divisor problems for second-order recurrences and connect Chebyshev polynomials, twin/cyclotomic phenomena, and recombination to the arithmetic of prime divisors.

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.

Figures (1)

• Figure 1: Sets of prime divisors of $X, WX, VX, CX$.

Theorems & Definitions (55)

• Lemma 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• proof
• Lemma 6
• proof
• Proposition 7
• Proposition 8