"If Fn has the same number of factors as Pn, divide both sides by p_i. Since Fn / p_i must be an integer, Fn must contain p_i, or else one of its factors f_i actually isn't prime by Euclid's Lemma."
You would have to prove that first. https://en.m.wikipedia.org/wiki/Euclid%27s_lemma:
"This property is the key[4] in the proof of the fundamental theorem of arithmetic
[4] In general, to show that a domain is a unique factorization domain, it suffices to prove Euclid's lemma and ACCP."
(https://en.m.wikipedia.org/wiki/Ascending_chain_condition_on... looks more intimidating than Euclid's lemma to me, but may be easier to prove.)