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"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.)

Got it. I was trying to do a proof while avoiding abstract algebra as much as possible, given how rusty I am with it.

Thanks for the feedback, and additional things to take a look at!

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