Erdős Problem #646 — Even Prime-Adic Valuations in Factorials

For every finite set $S$ of natural numbers consisting entirely of primes, there are infinitely many $n\in\mathbb N$ such that, for every $p\in S$, the $p$-adic valuation of $n!$ is even:

$$\forall p\in S,\qquad v_p(n!)\equiv0\pmod 2.$$