# 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) Nov 4, 2017 at 13:15

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.
• @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. Nov 4, 2017 at 14:36