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arXiv · 2607.09793

A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers

Josué Alexander Ibarra
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Abstract

An integer $n \ge 1$ is cyclic if $\gcd(n,\varphi(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_\sigma(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_\sigma$ is subadditive, $C_\sigma(m+n) \le C_\sigma(m)+C_\sigma(n)$ 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 $m=31$, $n=3928$, $C_\sigma(3959)=697 > 696 = C_\sigma(31)+C_\sigma(3928)$. The argument is short, and is verified by the Lean 4 kernel.

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Josué Alexander Ibarra. 2026-07-09. A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers. https://arxiv.org/abs/2607.09793

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