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

Abstract

Let $n\in {\mathbb{Z}}_{⩾2}$. 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 $n\ne m^2-1$ for $m\in {\mathbb{Z}}_{⩾2}$. 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 $n_1\sim n_2⟺P(n_1)=P(n_2)$. 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.

10 pages. v2: take into account suggestions by the referee. Accepted by J Exp. Math