Advanced Math/Use of Calculus in Number Theory
QUESTION: I'm doing a summer mathematics program at a local university in number theory. As I have just taken AP Calculus, I'm wondering whether my newly attained course material is applicable.
Are you "allowed" to use calculus and calculus theorems in number-theoretic proofs? Is there something technically wrong, or are you just not allowed to use different axioms and theorems in different fields of mathematics? Is it because calculus proofs often rely on number-theoretic ones? When one is proving basic arithmetical properties, such as the distributive or zero-product properties, are you disallowed from using calculus theorems on the basis that calculus presupposes those arithmetical properties?
Sorry if I asked too many questions! I haven't had a ton of experience in proofs, so I want to know sooner rather than later what is tacitly allowable under proofs.
ANSWER: Of course you are allowed -- many results are proved this way. There is an entire field of mathematics called "analytic number theory" which does precisely that. (I won't be foolish and send you a link to Wikipedia, I'm sure you can look it up on your own -- there are hundreds of textbooks and thousands of websites about analytic number theory.)
For example, the prime number theorem
is a result proven using analytic number theory
That being said, you should start from the bottom and work your way up. Most results in a first course in number theory will be "elementary," meaning they don't draw on other fields of mathematics.
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QUESTION: Thanks for the answert!
Quick follow-up questions: Are you always allowed to draw from different theorems and concepts in different fields of mathematics? Are those theorems and concepts universally applicable, or do some mathematical theories not work with some others (from an axiomatic or theoretical viewpoint)?
Interdisciplinary is common -- most active areas of mathematics research are not completely elementary, and at least span a few different subject areas (although many are closely related, e.g. geometry and topology). There is, most certainly, no rule against drawing from other areas of mathematics -- this often leads to important connections and helps use results from one area to help those in another.
To be somewhat candid, most mathematicians don't think about or care about the underlying axioms and so forth when they do their work (unless their work is in the area of studying such things). Sometimes certain work in mathematics will require invoking some technical aspect of the axiomatic foundations of mathematics, at which point it is more important to figure out whether the previous concepts and theorems are valid under a particular set of axioms or assumptions. But usually, it is not especially relevant so most working mathematicians don't really think too much about it.