Prime numbers and dynamics of the polynomial $x^2-1$

TL;DR

This paper investigates how prime divisors of certain polynomial-generated sequences relate to the original number, revealing infinite equivalence classes and partial separations among positive integers, with implications for algebraic structures.

Contribution

It introduces the concept of prime divisor sets for polynomial sequences and demonstrates their role in classifying integers, including infinite classes and partial separations.

Findings

Sets P(n) generate infinitely many equivalence classes.

Sets P(n) separate all positive integers up to 2^{29}.

Heuristics suggest P(n) may determine n uniquely.

Abstract

Let $n \in \mathbb{Z}_{\geqslant 2}$. By $P(n)$ we denote the set of all prime divisors of the integers in the sequence $n, n^2-1, (n^2-1)^2-1, \dots$. We ask whether the set $P(n)$ determines $n$ uniquely under the assumption that $n \neq m^2-1$ for $m \in \mathbb{Z}_{\geqslant 2}$. This problem originates in the structure theory of infinite-dimensional Lie algebras. We show that the sets $P(n)$ generate infinitely many equivalence classes of positive integers under the equivalence relation $n_1 \sim n_2 \iff P(n_1) = P(n_2)$. We also prove that the sets $P(n)$ separate all positive integers up to $2^{29}$, and we provide some heuristics on why the answer to our question should be positive.