What is the extent of the role of prime numbers in mathematics? I understand that prime numbers are called the "atoms" of number theory, because of the fundamental theorem of arithmetic. 
I also understand that they have some important practical applications such as in cryptography.
What I am wondering is, how important are prime numbers in mathematics in general?
In other words, if we didn't know anything about prime numbers, which areas of mathematics would break down? To what extent do proofs in the various fields of mathematics depend on prime numbers?
Edit: just to clarify a bit: I know that prime numbers "show up" in proofs sometimes, but I am trying to find out the extent of their importance in proofs. e.g. if abstract algebra in general is based on some theorem that completely relies on the existence of prime numbers, that would be very serious, but if just a couple of minor theorems do, or if the proofs can also be done without prime numbers, then that would mean their role is not so important.
 A: I will leave aside number theory, for which the question is too easy to answer.
In finite group theory primes are ubiquitous: Sylow subgroups and more generally $p$-groups have an important place. Among compact groups, $p$-groups generalize to pro-$p$ groups. 
In commutative algebra one of the most important notions is a prime ideal, which is both a generalization of prime numbers and an analogue of points in geometry. The role of prime ideals in proofs (e.g., localization at a prime ideal) is pervasive. Since commutative algebra is part of the technical apparatus of algebraic geometry, this makes prime ideals essential there as well. 
In hyperbolic geometry the prime geodesics are analogues of prime numbers. 
For Riemannian manifolds of negative curvature, analogues of the zeta-function of a number field and Artin $L$-function of a representation of a Galois group, both defined as Euler products over prime ideals, were used by Sunada in work on the length spectrum and are defined as products over prime geodesics.
In topology, (co)homology with coefficients in $\mathbf Z/p\mathbf Z$ for prime $p$ are important. Algebraic topologists use $p$-adic numbers. The Hilbert-Smith conjecture about transformation groups (an open version of Hilbert's fifth problem) is reduced to the special case of it for the additive group of $p$-adic integers.
In analysis, $L^2$-spaces play a central role and $2$ is a prime number...
