Has Gödel's first incompleteness theorem already led to theoretical ideas about number?

Gödel's first incompleteness theorem is based on the construction of a formula – a so-called Gödel formula – uttering its own flawless. Has such a formula already led to new knowledge in number theory (in addition to the incompleteness of arithmetic)?

Intuitively, a Gödel formula would be an excellent candidate as an additional axiom which, even if the axiomatization of arithmetic is not complete, would most likely result in new and interesting theorems. .

To my knowledge, this path has not been followed yet. Am I right or were the theoretical numbers results obtained by exploiting a Gödel formula?