Golden-ratio growth of Conway's subprime closure

Romain Popescu

math.NT Sep 12, 2026 · v1

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TL;DR

A number-theory proof of golden-ratio growth for Conway's subprime closure sets was formally verified in Lean 4, with code on GitHub.

Abstract

Let $s(m)$ be the Conway subprime function define the binary operation on the natural numbers $x \circ y= s(x + y)$, and denote by $C_n$, $n \ge 0$, the sequence of subsets of natural numbers defined by $C_0 = \{1\}$, and $C_{n+1} = C_n \cup (C_n \circ C_n)$. We prove the conjecture by Caragiu, Vicol and Zaki that $$\lim_{n\to \infty} \frac{ | C_{n+1}| }{ | C_n |}= \frac{1+\sqrt{5}}{2}.$$ 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.

Problem

Caragiu, Vicol, and Zaki conjectured that the sets C_n built from Conway's subprime function grow at the golden ratio, i.e. |C_{n+1}|/|C_n| tends to φ. The task is to prove this limit and its refined asymptotics.

Approach

The maximum M_n is shown to be prime with a Fibonacci upper bound M_{n+1} ≤ M_n + M_{n-1}. An almost-all Goldbach representation theorem for nearly equal primes extends complete prime prefixes at an almost-Fibonacci scale. Exponential-sum estimates over primes eliminate a gap between two asymptotic constants, forcing them equal. The proof was assisted algorithmically and its correctness formally verified in the Lean 4 proof assistant.

Results

The authors prove M_n ∼ Q_n ∼ cφ^n and |C_n| ∼ cφ^{n-1}, giving lim |C_{n+1}|/|C_n| = φ and lim |C_n|/M_n = 1/φ, confirming the conjecture.

Papers With

Golden-ratio growth of Conway's subprime closure