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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.

user56834
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  • They are quite important in algebra, especially group theory (you may know that, for example, $\mathbb Z_n$ is a field if and only if $n$ is prime), and their importance.

    Primes also crop up in analysis, most prominently in how they are connected to the Riemann zeta function.

     – 5xum Sep 08 '16 at 12:17
  • The fundamental theorem of arithmetic (aka unique-prime-factorization theorem) would not exist, which means - I guess - that the entire field of arithmetic would not exist. – barak manos Sep 08 '16 at 12:17
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    barak, could you explain what you mean with "the entire field of arithmetic would not exist"? people would still be able to add, subtract, multiply, divise, so arithmetic would still exist. – user56834 Sep 08 '16 at 12:23
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    I would say, @5xum, that your first example is more field theory than group theory. I'd go on to say that the only finite fields are of prime power order, and that the characteristic of a field is either a prime or zero, and therefore as soon as you get away from fields of characteristic zero, prime numbers are of crucial importance in field theory. – Gerry Myerson Sep 08 '16 at 12:33
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    I would also note that Godel's proof of the Incompleteness Theorem in logic depended on "Godel numbering", which in turn depended on primes (although that was just one convenient way of proving the theorem – I think one could put together a proof with no primes in it). – Gerry Myerson Sep 08 '16 at 12:35
  • It is perhaps worth noting that the prime numbers are often considered intrinsically interesting, and therefore exploring questions about prime numbers drives much mathematical research. So they're also important in the sense that they motivate math, as well as being technically useful in pursuit of other mathematical explorations. – Neal Sep 08 '16 at 13:52
  • If you want to solve Sudoku, prime numbers could help: http://www.acsu.buffalo.edu/~insrisg/LINKS/SudokuMath.pdf – Mr. Brooks Sep 08 '16 at 21:08

1 Answers1

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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...

KCd
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    The last one made me chuckle. – Nefertiti Sep 08 '16 at 14:47
  • @Nefertiti, I was kind of desperate to describe a role for prime numbers in analysis proper (not from analytic number theory) and that's the first thing I thought of. – KCd Sep 09 '16 at 01:45
  • OK, I interpreted the ellipsis as meaning that you had your tongue in cheek for that entry. Sorry if my comment was rude. Does the primeness of 2 explain the importance of $L^2$-spaces in some way? – Nefertiti Sep 09 '16 at 08:39
  • Here's another idea, maybe a bit of a stretch. Prime numbers are also generalised by maximal ideals of a ring. The set of maximal ideals (the spectrum) of a commutative unital Banach algebra over $\mathbf C$ is naturally a compact Hausdorff topological space.The Gel'fand representation theorem says that every commutative $C^*$-algebra is isomorphic to the algebra of continuous functions on its spectrum. – Nefertiti Sep 09 '16 at 08:49
  • I did indeed mean the last example just as a joke. There is no basis for primality of 2 having a role in $L^2$-spaces. You just want $1/p = 1/q$ when $1/p + 1/q = 1$. – KCd Sep 09 '16 at 16:15