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In a recent conversation with a friend, I was discussing the fact that out of all of the fields of math, number theory seems to be among those that apply ideas from a large number of different fields. For example, modern results in number theory incorporate complex analysis, algebraic geometry, lots of algebra, etc. In this sense, I see number theory as a more "applied" field (not in the usual sense of applied math), than say, something like algebra, hard analysis, algebraic geometry, or logic/set theory.

I was curious to what extent this can go the other way: in particular, what are some examples of theorems in other fields of pure math where a modern result in number theory was an important part of the proof?

Note: I'm not talking about elementary number theory. Of course, basic properties of the natural numbers are ubiquitous, and things like modular arithmetic, prime factorizations, the Chinese Remainder Theorem, etc. are indispensable in other fields. I also am looking for examples in other parts of pure math, not something like cryptography or computer science.

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I think algebraic geometry is the same. It uses everything from algebra, topology, analysis, and even some logic (model theory). – William Aug 3 '14 at 0:58
At the advance level, what exactly is the difference between algebra and number theory? – user150396 Aug 3 '14 at 1:02
@William I think you're right. I guess what I meant is that algebraic geometry has a foundational aspect that makes it useful in other fields. – Dorebell Aug 3 '14 at 1:07
Have you seen this question and its answers? – user160609 Aug 3 '14 at 1:48
Physics. There is a lot of modular forms in superstring theory. Basically, because the causal structure is determined up to conformal equivalence, so you get moduli spaces when quotienting. If you want more information see Witten's work. And of course, there is the Langlands program that is in some way equivalent to S-duality. – user40276 Aug 3 '14 at 2:59

I attended a talk where a mathematician turned biologist was making the claim that genetic codes could be analyzed using p-adic numbers. Specifically he was considering explaining amino acid attraction patterns using 5-adic anaylsis. The whole thing seemed like quite a stretch though to be honest.

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cooool. Biology will be to the 21st century as physics was to the 20th. It will be where the great scientific advances are. (Unless, of course, great scientific advances should turn out to be unpredictable.) – Michael Hardy Aug 3 '14 at 3:42

Möbius inversion is used to count irreducible polynomials over finite fields.

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Some number theory, like the theory of elliptic curves, has been applied to cryptography. Also coding theory may be mentioned. Furthermore there are many connections between number theory and string theory. Wave phenomena in quantum mechanics and quantum chaos also applies, and graph theory, especially expander graphs (Margulis) and related spectral theory.

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