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Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for suitable $n$, or perhaps other finite rings (such as quotients of extensions of $\mathbb{Z}$). Such results can be treated without a worry by a finitist. It is ingredients like this that are sufficient for, say, implementation of the RSA encryption scheme. However, at some point number theoretic results became statements (much more powerful and interesting statements!) about things like rings of adeles.

My question could be taken two ways:

What are the most powerful results in number theory which one can state and prove using the machinery of finite rings?

or

At what point in time did number theory move from considering finite rings to more analytic objects, and by what time was this move 'complete'?

For a reference post, one can point to the Lasker–Noether theorem (1921), which can be taken to be a statement about finitistic objects (and ignoring the possibility that infinite objects exist), namely finite modules $M$ for a ring $R$ (possibly infinite, in which case one could think of it as a finite ring by quotienting by the kernel of $R\to End(M)$).

My motivation for asking about this question is not to be controversial or obtuse, but to get an idea about how far a finitist might get in proving theorems in number theory. Much as reverse mathematics finds the precise strength of a subsystem of second-order arithmetic that is necessary to prove an analytic result (for example, the intermediate value theorem, the Heine–Borel theorem or the Bolzano–Weierstrass theorem), one could try to see how strong theorems in classical number theory are. This of course is something far outside the scope of a single MO question.

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The infinitude of primes is a pretty basic part of number theory that was found quite early in its development, so I don't understand what you mean by "at what point in time" did number theory not consider finite rings, because non-finite aspects have been around since the beginning. –  KConrad Dec 10 '12 at 4:09
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Dear David, Much about the adeles is actually finitistic; for example the Chinese remainder theorem, and the pigeon hole principle, are at the basis of many basic statements about the adeles; the adeles just provide a convenient way to encode the relevant information. My own view (or perhaps it is more of a hope/faith) is that most, even all, of the number theory that I am interested in is finitistic at its core. Cheers, Matthew –  Emerton Dec 10 '12 at 4:10
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I think the main interest in using infinitary methods is that the proofs can be much shorter. In other words, the use infinitary methods can be motivated for purely practical reasons. See my answer here: mathoverflow.net/questions/61632/… –  François G. Dorais Dec 10 '12 at 4:39
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Euclid's proof of the infinitude of primes does not use infinitary methods at all. It's a bit confounding because it is a negative statement: "infinite" = "not finite". The usual proof that $\sqrt{2}$ is irrational (= "not rational") has the same character but clearly does not use infinitary methods. See also this great post by Andrej Bauer on negative statements and their proofs - math.andrej.com/2010/03/29/… –  François G. Dorais Dec 10 '12 at 5:30
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Dear David, I find your question potentially very interesting, but right now, too ill-formulated to have a satisfying answer. First there are two questions, that you present as essentially the same, but that looks completely different to me. One is about when number theory stopped being finitist, the other is when number theory began using analytic tools. Both are not exactly well-defined, but they are very different. Let me focus on the first one, which is the one I find more interesting. The problem is in the definition of finitist. If one uses a too restrictive definition, then number.. –  Joël Dec 10 '12 at 19:47
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1 Answer

Dirichlet's Theorem on Primes in Arithmetic Progressions, proved in 1837, needing real-analytic methods could possibly be the first major candidate for a number-theoretic result departing from finite methods. (This was proved 50 years earlier than Prime Number Theorem).

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For an unusually large value of 50. –  Gerry Myerson Dec 10 '12 at 4:38
    
Although there is an elementary proof by Selberg: jstor.org/stable/1969454, which doesn't seem to use more than quadratic reciprocity. –  David Roberts Dec 10 '12 at 5:05
    
...and finite sums of logarithms. –  David Roberts Dec 10 '12 at 5:05
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Dirichlet did not have complex analysis at his disposal - all his arguments were real. Euler gave heuristic arguments suggesting that the sum of the reciprocal primes p less than x grow as log log x. –  Franz Lemmermeyer Dec 10 '12 at 18:07
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@ Benjamin Sternberg & Franz Lemmermeyer: Yes, the very term 'Euler product' connecting (real) zeta function with an infinite product running over primes suggests analysis in number theory had entered before Dirichlet. (So my reputation points can be brought down for the inaccurate answer I gave earlier). But we see mostly Riemann's name (as opposed to Euler) as showing the connection between distribution of prime numbers and Zeta function. Would be happy to get clarified on this point by the community here . –  P Vanchinathan Dec 11 '12 at 0:38
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