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

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 $C_σ(N)$ 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.