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, n^2 - 1, (n^2 - 1)^2 - 1,... . We ask whether the set P(n) determines n uniquely under the assumption that n ≠ m^2 - 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 n_1 ∼ n_2 ⇔ P(n_1) = P(n_2). 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.
Fields of Study & Subjects
Analytic Number Theory Research
Algebraic Geometry and Number Theory
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Journal of Experimental Mathematics
- Volume
- 2
- Issue
- 1
- Pages
- 194–204
- Publication date
- 1 March 2026
- Language
- EN