AI 中文总结
研究针对科恩关于索菲热尔曼循环数的次可加性猜想，通过给出 \(m = 31\)，\(n = 3928\) 时 \(C_\sigma(3959) > C_\sigma(31) + C_\sigma(3928)\) 的反例进行反驳，论证简短且经Lean 4内核验证。
AI 中文摘要
整数 \(n \geq 1\) 若满足 \(\gcd(n,\varphi(n)) = 1\)（即每个阶为 \(n\) 的群都是循环群）则是循环的，若 \(n\) 和 \(2n + 1\) 都是循环的则是索菲热尔曼循环的。设 \(C_\sigma(N)\) 统计 \([1,N]\) 中的索菲热尔曼循环整数。科恩猜想 \(C_\sigma\) 是次可加的，即 \(C_\sigma(m + n) \leq C_\sigma(m) + C_\sigma(n)\) 对所有 \(1 \leq m \leq n\) 成立，他在检查 \(m,n \leq 10^6\) 时未找到反例。我们给出一个反例：当 \(m = 31\)，\(n = 3928\) 时，\(C_\sigma(3959) = 697 > 696 = C_\sigma(31) + C_\sigma(3928)\)。论证简短且经Lean 4内核验证。
英文摘要
An integer $n \ge 1$ is cyclic if $\gcd(n,φ(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_σ(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_σ$ is subadditive, $C_σ(m+n) \le C_σ(m)+C_σ(n)$ for all $1 \le m \le n$ (his Conjecture 66), having checked $m,n \le 10^6$ without finding a counterexample. We give one: at $m=31$, $n=3928$, $C_σ(3959)=697 > 696 = C_σ(31)+C_σ(3928)$. The argument is short, and is verified by the Lean 4 kernel.
Comments3 pages. Ancillary file: Lean 4 formalization (anc/cohen66.lean); the refutation is Lean kernel-checked, depending only on propext, Classical.choice, and Quot.sound
