The weighted vector-composition prime criterion

Let $N\geq 1$, let $\mathbb{N}$ denote the nonnegative integers, and let $\mathbf{1}=(1,\ldots,1)\in\mathbb{N}^N$. For $\mathbf{s}=(s_1,\ldots,s_N)\in\mathbb{N}^N$, define

$$U_{\mathbf{0}}=\{\mathbf{s}\in\mathbb{N}^N\mid s_i\in\{0,1\}\}.$$

Let $\binom{m}{\mathbf{k}}_f$ denote the $f$-weighted vector-composition coefficient. Weighted vector-composition prime criterion. If $f(\mathbf{x})=1$ for every $\mathbf{x}\in U_{\mathbf{0}}$, then an integer $q>1$ is prime if and only if $m$ divides

$$\binom{m}{q\mathbf{1}-2m\mathbf{1}}_f$$

for every integer $m$ satisfying $0\leq 2m\leq q$. This proposed criterion extends the Mann–Shanks prime criterion from integer compositions to weighted vector compositions; its status is not established in the supplied text.