Prime Numbers and Dynamics of the Polynomial x² - 1

Abstract

Let n ∈ Z_>=2. By P(n) we denote the set of all prime divisors of the integers in the sequence  n,  n² - 1,  (n² - 1)² - 1,... . We ask whether the set P(n) determines n uniquely under the assumption that n ≠ m² - 1 for m ∈ 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 n₁ ∼ n₂ ⇔ P(n₁) = P(n₂). We also prove that the sets P(n) separate all positive integers up to 229, and we provide some heuristics on why the answer to our question should be positive.