2502.11929
PRIME NUMBERS AND DYNAMICS OF THE POLYNOMIAL x2 − 1
Ivan Penkov, Michael Stoll
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves there are infinitely many equivalence classes under n1 ∼ n2 ⇔ P(n1)=P(n2) by exhibiting that for primes p ≡ ±3 (mod 8), the sets P(p) are pairwise distinct (via Lemma 4.1 and Theorem 4.2 in Section 4) . The candidate model independently proves the same claim by a constructive CRT-based prescription using primes p for which x^2−x−1 has a root modulo p (i.e., 5 is a quadratic residue), allowing inclusion/exclusion of each such prime in P(n). Both arguments are correct and reach the same conclusion with different methods. The paper’s broader context and computational evidence (Sections 1 and 3) are consistent with the model’s local congruence control perspective .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
Section 4’s proof that there are infinitely many classes is sound and elegantly elementary. It complements the paper’s broader computational and heuristic narrative on uniqueness. Minor clarifications (e.g., explicitly linking fixed points modulo p to exclusion from P(n), and making some standard quadratic residue facts explicit) would aid readability and cross-linking with the heuristic Section 5.