# Subfields of Number Theory that aren't about prime numbers.

A friend of mine who knows more mathematics that I do, told me that nowadays number theory is basically about studying prime numbers.

I'm under the impression that solving diophantine equations and cryptography for example are subfields of number theory.

I'm under the impression that although at first glance these subfields dont seem to have anything to do with prime numbers, prime numbers play a central role in them. (is this true?)

So it seems that in certain subfields of number theory that aren't directly about prime numbers, the researchers study and tackle their problems by studying prime numbers, and so prime numbers would be central in such subfields.

So I'm asking for examples of subfields of number theory that don't rely heavily/aren't prime number focused.

Thanks.

• You can search about multiplicative vs additive number theory (even if the latter includes things like Goldbach conjecture) – reuns Nov 4 '17 at 13:15

## 2 Answers

As you said, arithmetic geometry, which is a part of modern number theory, is not directly focused on prime numbers. It is about algebraic geometry applied to Diophantine problems and other things. For example, Grothendieck's anabelian geometry is about schemes, or algebraic varieties over a field $K$, with an algebraic fundamental group (etale fundamental group), which is a profinite group. This unifies Galois theory and the theory of covers form topology. Mochizuki is using many techniques for "his proof of the ABC-conjecture" which go far beyond "prime numbers". On the other hand, the $p$-adic worlds here is extremely important, and in several ways, prime numbers are always present.

• I admit I am very unfamiliar with arithmetic geometry and ABC conjecture. Yet searching ABC conjecture reveals the words co-prime in the problem statement, and so it seems that problem is (very?) prime-number related. As for arithmetic geometry. You mentioned it's about algebraic geometry applied to Diophantine problems and other things. So there problems in this area that have nothing to do with prime numbers? Thanks. – trynalearn Nov 4 '17 at 13:37
• "nothing to do with prime numbers" is impossible in number theory. But as the word "algebraic geometry" already says, this has a focus on other things. It would be inappropriate to say that arithmetic geometry is mainly about prime numbers. – Dietrich Burde Nov 4 '17 at 15:10

As an approximation, analytic number theory is largely to do with studying the prime numbers, but algebraic number theory is not, for the most part. Prime numbers (and prime ideals) are hugely important in algebraic number theory, but the field is not primarily concerned with things like the distribution of the prime numbers and which particular numbers are prime.

As an example, Fermat's Last Theorem was proved using techniques of algebraic number theory. The weak Goldbach conjecture was proved using analytic number theory.

• Aha. So within algebraic number theory are there a number of problems that have nothing to do with prime numbers? I know the phrase 'nothing to do' can be problematic since the theory of "things that aren't prime numbers" may be very useful in answering things about primes. Examining the statement of Fermat's Last Theorem it seems to have nothing to do with prime numbers. Could this be an example of such a problem? Or do the known proofs depend upon prime number theory? – trynalearn Nov 4 '17 at 13:42
• @trynalearn $a^n +b^n = c^n$ reduces to $a^p +b^p =c^p$ but otherwise it doesn't look about prime numbers. Except that $a^p -c^p= \prod_{m=1}^p(a-\zeta^m c) \in \mathbb{Z}[\zeta]$ where $\zeta = e^{2i \pi /p}$ and it factorizes more in a product of prime ideals. This method didn't work to answer FLT but it motivated a lot of research (by Kummer) on (abelian) number fields. Finally Wiles proof of FLT comes from the theory of L-functions of elliptic curves, modular forms and Galois representations where prime numbers are everywhere. – reuns Nov 4 '17 at 14:36
• @trynalearn The statement itself has nothing to do with prime numbers. Essentially, the proof depends (roughly) on the fact that every whole number has a unique factorization into prime numbers, but this is a property of whole numbers in general, not of the prime numbers themselves. There is an anecdote in which Grothendieck (who is a giant in algebraic geometry, and by extension in algebraic number theory) was asked to name a prime number. 'OK,' he said. 'Take 57.' – John Gowers Nov 4 '17 at 15:17