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The mathematical field of tackling number theory problems in this way is called analytic number theory.

https://en.wikipedia.org/wiki/Analytic_number_theory

The prime number theorem, on how prime numbers are distributed amongst the integers, was first proved using analytic techniques.

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Analytic number theory exists and involves calculus, but it's not what the linked post is about. The article talks about Hensel's lemma, which is a purely algebraic statement with a purely algebraic proof, which, however, is inspired by techniques from calculus. This is typically still categorized as algebraic number theory.

Get a load of number theorists in a room and there will always be a fight between the analytic and algebraic.

Hensel's lemma is an analytic fact about the radius and speed of convergence of Newtons method in the p-adics.

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