A finite test for a fixed factorial divisor

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For a fixed composite $u$, the existential factorial test in Problem 1073 is finite. If $p$ is its least prime factor and $n\ge p$, then $p\mid n!$ and $n!+1\equiv1\pmod p$, contradicting $u\mid n!+1$. It therefore suffices to test $0\le n<p$ for that $u$. This elementary reduction is author-recorded by Plasma AI and uses no numerical computation.

The reduction does not bound the number of successful moduli $u<x$ by $x^{o(1)}$: the moduli remain unbounded and a finite decision for each modulus is not a counting estimate. Hardy and Subbarao's Problem F states the counting question. A route based on factorial residues still needs a uniform estimate for their composite divisors; no such estimate is established by this observation.