# Where is number theory used in the rest of mathematics?

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.

• 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. Commented Mar 9, 2012 at 15:05
• 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. Commented Mar 9, 2012 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. Commented Mar 9, 2012 at 17:37
• I think this answer to the analogous question in algebraic geometry is also applicable here: mathoverflow.net/questions/77195/… Commented Mar 10, 2012 at 19:01
• I would highly recommend Minhyong Kim's essay "Why everyone should know number theory" people.maths.ox.ac.uk/kimm/lectures/numbers.pdf Commented Jul 22, 2020 at 18:39

## 38 Answers

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.

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.)

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.

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.

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.

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.

Excerpt from "Moonshine Link Discovered for Pariah Symmetries" (2017)

Apart from their cameo role in the classification of finite symmetries, the pariahs “have not appeared anywhere in mathematics,” wrote Ken Ono, a mathematician at Emory University, in an email. “They are something like the super heavy metals in the periodic table of elements.”

Now Ono, John Duncan of Emory, and Michael Mertens of the University of Cologne in Germany have succeeded in welcoming one of the pariahs, called the O’Nan group, into the framework of a theory known as “moonshine.” Originally developed decades ago for a gargantuan symmetry structure called the monster group, moonshine forges deep connections between groups of symmetries, models of string theory and objects from number theory called modular forms.

The new work, which the researchers describe today in Nature Communications, puts the O’Nan group at the crest of this new wave of moonshine, which links certain symmetry groups to special classes of “weight 3/2” modular forms, objects that also show up in natural counting functions for black holes and for higher-dimensional generalizations of strings called “branes.”

(See the Nature article Pariah Moonshine.)

Edit: Jan 10, 2021:

In addition see the MO-Q "The Dedekind eta function in physics" in which I link to a lecture by Cheng and Felder that sketches very nicely the relationships among string theory, moonshine, J-homomorphism, and modular forms, such as the Dedekind eta function.

• Potentially confusing point: Nature Communications is a separate journal from Nature, but part of the Nature group. Commented Aug 31, 2019 at 3:05
• For mathematical physics/quantum field theory, see "Numbers and functions in quantum field theory" by Schnetz arxiv.org/abs/1606.08598 Commented Sep 3, 2019 at 2:47
• Sanders and Wang, "Number theory and the symmetry classification of integrable systems" Commented Sep 5, 2019 at 20:42

There are some very interesting examples of pointsets with pure point diffraction in Aperiodic order, coming from number theory.

The simplest example is the set of square free integers $$S= \{ n \in \mathbb Z: \forall p \mbox{ prime }, p^2 \nmid n \}$$

The issue for diffraction is that for this example it matters how one averages when calculating the diffraction. It was observed in Arxiv paper that if one uses the "natural" averaging sequence $$A_n=[-n,n]$$ the diffraction is pure point and can be calculated explicitly. The proof is basically a long technical computation using the Chinese Remaining Theorem.

Note here that this is the diffraction of the square of the Mobius function.

In recent years this example motivated some progress in the area of Aperiodic Order, leading to a systematic study of maximal density weak model sets and $$\mathcal{B}$$-free systems. As a side note, the primes actually fit nicely in this picture, but their density and hence diffraction is 0, so all results about primes coming from this direction are trivial.