There are so many applications of the Axiom of Choice (and consequently its equivalents) in number theory. But do you know any application of the Zorn's Lemma in Number Theory !? I mean a theorem or a problem in this field that uses Zorn's Lemma ?
Edit: of course any application of AC is equivalent to an application of the Zorn's Lemma. But I am sure you agree that some proofs work better with AC, some work better with the Well ordering axiom and some work better with Zorn's Lemma. When we prove that any vector space has a base we use Zorn's lemma. Nobody writes a proof with well ordering axiom while anybody can write such a proof.