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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.11929 |
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| _version_ | 1866911258784038912 |
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| author | Penkov, Ivan Stoll, Michael |
| author_facet | Penkov, Ivan Stoll, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime numbers and dynamics of the polynomial $x^2-1$ Penkov, Ivan Stoll, Michael Number Theory Dynamical Systems 37P05, 37P15, 15B30, 17B45, 11Y99 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. |
| title | Prime numbers and dynamics of the polynomial $x^2-1$ |
| topic | Number Theory Dynamical Systems 37P05, 37P15, 15B30, 17B45, 11Y99 |
| url | https://arxiv.org/abs/2502.11929 |