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There is a proof of the infinitude of primes using topology. I was only informed of the existence of this proof. They say it's very elegant. One could show how this proof?

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The proof is presented in Wikipedia: en.wikipedia.org/wiki/… – Brad Dec 20 '11 at 16:00
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If you want to see a lengthy discussion on the originality and interest of this proof, head over to the comments at mathoverflow.net/questions/34699/… – David Speyer Dec 20 '11 at 17:53

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You may also like the paper On the exotic topology of the integers by Mezö and Lovas: Fürstenberg's topology turns $\mathbb Z$ into a metrizable, totally disconnected space, and $(\mathbb Z,+,\cdot)$ is a topological ring with respect to this topology (this last is in fact quite elementary!).

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Great read. Thanks! – Isaac Solomon Dec 21 '11 at 0:04
I'm using this paper in some research I'm finally getting back to after a year off for health reasons.It's definitely worth people noticing it. – Mathemagician1234 Mar 5 '12 at 5:30
up vote 16 down vote

Here is a variant of Fürstenberg's proof that does not use topological notions (which obscure the main idea): We are arguing about periodic subsets of ${\mathbb Z}$. The set of integers not divisible by $p$ is periodic for any $p>0$, and the intersection of two periodic sets is periodic. If there were only finitely many primes the set $\{-1,1\}$ would be periodic.

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It would help the reader to precisely define what "periodic" means above. – Gone Dec 20 '11 at 18:02
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I'd say that a subset of $\mathbb Z$ is periodic iff its characteristic function is. --- I find the answer brilliant! – Pierre-Yves Gaillard Dec 20 '11 at 18:14
I am afraid to make this last remark true you have to define the characteristic function of a set as: 0 for elements in the set, and 1 for elements not in the set. In other words, this is the dual of the common way of defining characteristic function. – boumol Dec 20 '11 at 19:49
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@Mathemagician1234 Please see my comment to Brian in my answer. – Gone Dec 20 '11 at 21:05
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@buomol: Why? Periodicity of $f$ is the same as periodicity of $1-f$. – sdcvvc Dec 20 '11 at 22:48
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Eliminating the (unneeded) topological language from Fürstenberg's proof shows that it is simply the following trivial variation on Euclid's proof. If there are only finitely many primes $\rm\:p_1,\ldots,p_n\:$ then there are infinitely many units $\rm\:1+p_1\:\cdots\:p_n\ \mathbb Z,\:$ contra $\:\mathbb Z\:$ has only finitely many units $\pm1\:.\:$

For a much less trivial reinterpretation of Euclid's proof see my fewunits generalization.

THEOREM $\ $ An infinite ring $\rm R$ has infinitely many max ideals if it has fewer units $\rm U = U(R)$ than it has elements, i.e. $\rm\:|U| < |R|$.

The marvelous thing about this proof is that it preserves the constructivity of Euclid's proof. The key idea is that Euclid's construction of a new prime generalizes from elements to ideals, i.e. given some maximal ideals $\rm P_1,\ldots,P_k$ then a simple pigeonhole argument employing $\rm CRT$ implies that $\rm 1 + P_1\cdots P_k$ contains a nonunit, which lies in some maximal ideal $\rm P$ which, by construction, is comaximal (so distinct) from the prior max ideals $\rm P_i\:.\:$ Follow the above link for full details.

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The objection in the first paragraph rather misses the real point of Fürstenberg’s proof, which is that topological language/ideas can offer a possibly productive way of thinking about something apparently non-topological. And the restatement in terms of units seems to me rather less perspicuous than the topological version. – Brian M. Scott Dec 20 '11 at 20:14
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@Brian Don't let the topological language deceive you. As stated F's result is only syntactically topological - not semantically. If you deem F's proof innately topological then you may as well deem any trivial use of Boolean algebra topological, since that's what it amounts to when examined closely. Syntactic reformulations are a dime a dozen without significant semantic consequences. Of course it goes without saying that such evaluations are highly subjective. But some other mathematicians agree, e.g. B. Conrad in said MO thread. – Gone Dec 20 '11 at 21:20
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@Adam: No, it isn’t. It’s a fundamentally different way of looking at the proof. And the fundamental idea $-$ of using topological ideas in a number-theoretic context $-$ was obviously not a dead end, however trivial this early exercise may have been. – Brian M. Scott Dec 20 '11 at 22:16
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@Brian Please do elaborate on the "different possibilities and different modes of thought" that this suggests to you and, more importantly, the interesting number theory resulting from such. There can be no doubt that some topological notions do play nontrivial conceptual roles in modern number theory (these ideas were known long before F's proof). The same cannot be said for F's trivial reformulation of Euclid's proof (as presented here) – Gone Dec 20 '11 at 22:35
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That was a general comment about the importance of language and mathematical context: you appeared to think that syntactic reformulations are necessarily insignificant, which is nonsense. With ‘not a dead end’ I was referring to the application of topological dynamics in ergodic Ramsey theory and combinatorial number theory generally, not to the specific topological elements of this pretty but elementary result (which I don’t consider a trivial reformulation of Euclid’s proof). – Brian M. Scott Dec 20 '11 at 23:07
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up vote 6 down vote

This is Fürstenberg's proof (follow the link). It is indeed beautiful.

PS: Oh, sorry, I didn’t see Brad’s answer before answering (in fact curiously I think it was not visible when I answered).

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