Talk:Furstenberg's proof of the infinitude of primes

Topology axioms
The verification of the axioms seems wrong and incomplete.


 * Any union of open sets is open: for any collection of open sets Ui and x in their union U, any of the numbers ai for which S(ai, x) ⊆ Ui also shows that S(ai, x) ⊆ U.

How does this show that any union of open sets is open? I don't see the relevance of the conclusion. Why not just say that because each nonempty open set is a union of arithmetic sequences, a union of open sets is also a union of arithmetic sequences?


 * The intersection of two (and hence finitely many) open sets is open: let U1 and U2 be open sets and let x ∈ U1 ∩ U2 (with numbers a1 and a2 establishing membership). Set a to be the lowest common multiple of a1 and a2. Then S(a, x) ⊆ S(ai, x) ⊆ Ui.

Again, what does this have to do with the topology axiom? It has not been shown that U1 ∩  U2 is a union of arithmetic sequences. This argument would make more sense if we were talking about bases instead of open sets. Using the base S(a, b) for any a,b, then the above argument does show that the intersection of any two sets in the base is another set in the base, which proves it is a valid base. But just talking about open sets and not bases, it does not make sense, because an open set is not necessarily represented by a single arithmetic sequence S(a, b).

I'm a novice to topology so I hesitate to change the article - can someone with more confidence check this and if necessary clarify or fix it?

Halberdo (talk) 16:00, 26 September 2016 (UTC)

Contact
I am interested in getting in contact with the person who created this page.

I am an undegraduate mathematics major. I discovered this proof around the same time this article was posted, and it motivated me to demand that my school offer a course in point-set topology. In March 2008, I gave a presentation on this proof to the meeting of the Southeastern Section of the Mathematical Association of America. It won an award there.

Anyway, this proof fascinates me. I am interested in hearing from people whom it fascinates as well.


 * I created the page after hearing about the proof last year from a fellow-mathematician friend of mine. He describes it as &ldquo;the ultimate rabbit-out-of-hat proof&rdquo;, i.e. you introduce some apparently irrelevant objects, observe a few properties, and something remarkable appears in a puff of magic. :-)  Sullivan.t.j (talk) 11:58, 17 June 2008 (UTC)

what's so great about this long and complex proof? Just say: If there were only finite primes, you could multiply them all together, add one, and get a finite number that, being relative prime to all of them, must have a prime factor not in "all primes" -- an absurdity.  —Preceding unsigned comment added by 82.124.209.97 (talk) 17:34, 4 November 2008 (UTC)
 * Yes, a chap named Euclid of Alexandria noticed this fact (that there are infinitely many primes) some time ago. Not satisfied with his approach, some sad people called mathematicians continued to hunt for new and inventive ways of proving what they already knew to be true, in the hope that this might be either enlightening or at least stave off boredom on cold winter nights!  ;-)  Sorry to be put it that way, but this is kind of the point:  the statement of the theorem is, of course, not new, but Furstenberg's approach is regarded as particularly inventive and representative of his ability to combine methods from apparently unrelated areas of mathematics and produce interesting insights and results. Sullivan.t.j (talk) 17:54, 4 November 2008 (UTC)

I am not the person who created this page but I know a little bit about the proof. The proof is interesting but I am not surprised. One can always establish a link between two different branches of mathematics. This is a connection between number theory and topology.

If you are interested in this proof, you may find the proof that every subgroup of a free group is free interesting (which uses in the proof several tools from algebraic topology!). What in particular did you want to know about the theorem?

Topology Expert (talk) 04:10, 5 November 2008 (UTC)

Proof By Contradiction
In many articles, it seems, including this one, Euclid's proof is stated as having been by contradiction. However, the way Euclid did it it was not. I wish I could link to a discussion page or something better to back this up, but meanwhile see Euclid's Theorem. The issue is a tricky one because the form in which Euclid's proof is usually given is, in fact, by contradiction, so when we say 'Euclid's proof' do we mean his version or the popular version? So I'm not changing it for now. It's a nitpicky point anyway. —Preceding unsigned comment added by 66.188.36.26 (talk) 20:53, 26 July 2009 (UTC)

It's just a detail, but I think it should be noted that a can't be 0. 84.194.94.70 (talk) 16:37, 7 March 2010 (UTC)
 * Thanks for noting this; it has now been corrected. Feel free to correct any mistakes you notice in Wikipedia articles, if possible. PS  T  11:03, 8 March 2010 (UTC)

I've just corrected it. This edit introduced the error. It's quite a widespread error, asserted in print by many respectable mathematicians. But nonetheless it's an error. Michael Hardy (talk) 17:40, 11 March 2010 (UTC)

Usual Euclidean Topology
There is no "usual Euclidean Topology" on the integers. The induced subspace structure is just the discrete topology, and his really no relation to the topology of R^n. It is impossible to compare two topologies with different underlying sets and the way it is written now there is some reference to some usual Euclidean topology on the integers which really does not exist. I am removing this reference for the above reasons. — Preceding unsigned comment added by 209.134.82.48 (talk) 05:55, 22 June 2013 (UTC)
 * The name "usual Euclidean Topology" probably meant that the topology is induced by the "usual Euclidean inner product" on the integers (we can define inner product on modules as Z^n). Anyway, the mention was useless, so it is good to suppress it. — Preceding unsigned comment added by 210.119.97.56 (talk) 06:12, 17 October 2013 (UTC)