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Golden-ratio growth of Conway's subprime closure
Abstract
Let s(m) be the Conway subprime function define the binary operation on the natural numbers x ∘ y= s(x + y), and denote by Cn, n ≥ 0, the sequence of subsets of natural numbers defined by C0 = {1}, and Cn+1 = Cn ∪ (Cn ∘ Cn). We prove the conjecture by Caragiu, Vicol and Zaki that limn→ ∞ ( | Cn+1| )/( | Cn |)= (1+√5)/2. The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.
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BibTeX
@misc{popescu2026goldenratio,
title = {{Golden-ratio growth of Conway's subprime closure}},
author = {Romain Popescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14188},
eprint = {2609.14188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14188},
}
RIS
TY - UNPB
TI - Golden-ratio growth of Conway's subprime closure
AU - Romain Popescu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14188
UR - https://arxiv.org/abs/2609.14188
AB - Let s(m) be the Conway subprime function define the binary operation on the natural numbers x ∘ y= s(x + y), and denote by C_n, n ≥ 0, the sequence of subsets of natural numbers defined by C_0 = {1}, and C_n+1 = C_n ∪ (C_n ∘ C_n). We prove the conjecture by Caragiu, Vicol and Zaki that lim_n→ ∞ ( | C_n+1| )/( | C_n |)= (1+√5)/2. The underlying mathematical proof in this paper was constructed with some algorithmic assistance from GPT-6 Astra and its correctness has been formally verified using the Lean 4 proof assistant.
ER -