There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fileds of mathematics. For example consider the Fermat's conjecture (Wiles's theorem), what are the consequences and impacts of the statement "for each $n\geq 3$ there are no natural numbers like $x,y,z$ such that $x^n+y^n=z^n$" on the other fields of mathematics? Of course the tools which Weils and others discovered to prove Fermat's conjecture have many applications in geometry and other realms but what about the result itself? My question is about these possible relations and interactions:

Question: Please introduce some references for any known theorem like the following statements:

1. If the conjecture C in number theory is true then the statement S in the field F of mathematics will be true.

2. If the statement S in the field F of mathematics is true then the the conjecture C in number theory will be true.

Remark 1: Particularly the following cases are more interesting for me:

C $=$ Schanuel's conjecture, Goldbach's conjecture, Fermat's conjecture (Weils's theorem).

F $=$ Logic, set theory, model theory.

Remark 2: Although it is not my main propose but by the answers of the question one can find some possible non-number theoretic ways to prove or refute conjectures in number theory.