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Where is number theory used in the rest of mathematics?

To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to answer them?

To put it another way still: imagine a mathematician with no interest in number theory for its own sake. What are some plausible situations where they might, nevertheless, need to learn or use some number theory?

Edit It was swiftly pointed out by Vladimir Dotsenko that the borders between number theory and algebraic geometry, and between number theory and algebra, are long and interesting. One could answer the question in many ways by naming features on that part of the mathematical landscape. But I'd be most interested in hearing about uses for number theory that aren't so obviously near the borders of the subject.

Background In my own work, I often find myself needing to learn bits and pieces of other parts of mathematics. For instance, I've recently needed to learn new bits of analysis, algebra, topology, dynamical systems, geometry, and combinatorics. But I've never found myself needing to learn any number theory.

This might very well just be a consequence of the work I do. I realized, though, that (independently of my own work) I knew of no good answer to the general question in the title. Number theory has such a long and glorious history, with so many spectacular achievements and famous results, that I thought answers should be easy to come by. So I was surprised that I couldn't think of much, and I look forward to other people's answers.

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This is too empty for an answer, so I'll just type a comment. 1) For your third interpretation I have at least one relevant experience of my own - "factor systems" (some) number theorists deal with when talking about central simple algebras over number fields did contribute to the development of homological algebra in general, and learning that somehow expanded my homological algebra horizons a bit. 2) Sometimes, it's hard to say where the border between number theory, commutative algebra, and algebraic geometry lies. Around that "triple point" there should be many potential answers. –  Vladimir Dotsenko Mar 9 '12 at 15:05
For instance Number Theory is used in Algebraic Geometry when studying problems in characteristic $p$. But even in characteristic 0, some algebraic varieties arise from arithmetic constructions. Hilbert modular surfaces or Shimura varieties are examples of this situation. Also, the recent classification of fake projective planes by Prasad & Yeung (2007) uses Number Theory in a crucial way. I think there are countless examples like these –  Francesco Polizzi Mar 9 '12 at 15:06
Also the study of the Ring of Endomorphisms for abelian varieties needs the understanding of some Number Theory, especially number fields and quaternion algebras. –  Francesco Polizzi Mar 9 '12 at 15:09
Thanks very much for the comments. Can I suggest that people write this kind of thing as answers rather than comments, though? That way, replies to your comments are organized more neatly. –  Tom Leinster Mar 9 '12 at 15:11
Number theory is used in the representation theory of finite groups to address rationality questions. Algebraic integrality seems to come up just about everywhere. –  Grant Rotskoff Mar 9 '12 at 17:37

36 Answers 36

Elliptic curves are the basis for a type of public key cryptography known as elliptic curve cryptography.

Also I attended a colloquium recently that talked about the applications of elliptic curves to string theory.

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The uniqueness of the finite Ree groups of type $^2G_2$ was established by E. Bombieri using extremely tricky number theoretical methods (involving involved elimination methods). As Stephen D. Smith wrote in his review on MathSciNet:

This result has considerable importance in the classification of finite simple groups. Ordinary mortals such as the present reviewer are overawed by the author's tour de force.

(Bombieri, Enrico; Odlyzko, A.; Hunt, D. Thompson's problem (σ2=3). Appendices by A. Odlyzko and D. Hunt, Invent. Math. 58 (1980), no. 1, 77–100.)

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It also finds use in the theory of information theory and algorithms (which are essentially mathematical topics); for example, number theoretic transforms are arguably the neatest way of achieving the optimal known complexity for multiplying arbitrarily long integers on a classical computer. Other topics also involving some amount of number theory are hashing, random number generation and error detection & correction.

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Well, the Fibonacci sequence belongs to number theory. Doesn't it ? Yet it is fundamental in many parts of Mathematics, at least in sorting algorithms.

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  1. Algebraic topology and theory of formal groups use arithmetic properties of binomial coefficients like $$d(n)=\left({n\choose1},{n\choose2}, \ldots, {n\choose n-1}\right)=\begin{cases} p,& \text{if } n=p^k;\\ 1,& \text{else};\\ \end{cases}$$ $$\left({n\choose 2},{n\choose 3}, \ldots, {n\choose n-2}\right)=d(n)d(n+1)$$ (and something more serious, see Hazewinkel, M. Formal groups and applications. 1978).

  2. Discrete integrable systems (and algebraic topology as well) use different kinds of special functions especially elliptic ones.

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This is maybe too elementary --- very simple number theory is used in the classification of finite Markov Chains.

See for instance: Karlin & Taylor: A First Course in Stochastic Processes.

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