Prime numbers and dynamics of the polynomial $x^2-1$
Ivan Penkov, Michael Stoll
TL;DR
This work investigates whether the prime-divisor sets $P(n)$, defined along the orbit $a_0=n$, $a_{k+1}=a_k^2-1$, determine $n$ uniquely. It combines rigorous results—proving $P(n)$ is infinite for all $n\ge2$ and establishing infinite and near-sparse behavior—with extensive computational evidence showing that $P(n)$ can separate all integers up to at least $2^{29}$ using primes up to $10^4$. The authors construct infinitely many equivalence classes under $n\sim m$ iff $P(n)=P(m)$ and provide heuristics, via backward-tree models and Chebotarev-type density, predicting the persistence of informative primes and suggesting a positive answer to the uniqueness question. They also outline speculative connections to deep conjectures (e.g., Vojta) and a potential strategy to deduce orbit relations from gcd growth, highlighting the intersection of arithmetic dynamics with Lie-theoretic structure in infinite-dimensional settings.
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.