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Which are the deepest theorems with the most elementary proofs?
I give two examples:
i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
ii) Proof that the halting problem is undecidable using diagonalization

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    This should be community wiki, I think.2011-10-22
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    Proving the incompleteness theorems requires a little more than just diagonalization.2011-10-22
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    Something like this got asked in MO IIRC. Now if only I could find it...2011-10-22
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    Isn't depth, by definition, inversely proportional to the elementariness of proofs?2011-10-22
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    How do you define the "deapness" of a proof or of a theorem?2011-10-22
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    If you read Spivak's "Calculus on Manifolds", he specifically structures the whole book around making Stokes' Theorem trivial to prove... does that count?2011-10-22
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    @detly I don't know the particular example to which you are referring, but I don't think it counts as "simple" if you tuck away all the hard work in lemma after lemma and use them produce a two-line proof of a big result.2011-10-23
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    I am sure this is not what you had in mind, but I am tempted to mention the "elementary" proofs of the Prime Number Theorem due to Erdős and Selberg. :)2011-10-23
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    possible duplicate of [Surprising Generalizations](http://math.stackexchange.com/questions/1352/surprising-generalizations)2011-10-23
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    Why is this closed? Reopen, thank you. Leave reason if you vote to close?2011-10-26
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    How about "reopen, please"?!? Also "PLZ reopen" in the title is obnoxious.2011-10-26
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    @GM2001: Please stop editing titles to contain messages like "AWESOME" or " REOPEN !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!...please!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!". This is not the first time you're doing it.2011-10-26
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    @GM2001: If you edit this question any more, I will lock it, which will prevent further edits, comments, and answers.2011-10-26

2 Answers 2

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These perhaps aren't particularly deep, but they are the first that come to mind.

  1. Irrationality of $\sqrt{2}$ by contradiction.
  2. Uncountability of the reals by diagonalization.
  3. Existence of graphs with arbitrarily high girth and chromatic number by the probabilistic method.
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    I think the linked proof in (1) is misnamed, and is not a proof by *infinite descent* (unlike [this one](http://en.wikipedia.org/wiki/Infinite_descent#Irrationality_of_.E2.88.9A2)).2011-10-22
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I think one should not confuse "important with "deep". The facts that $\sqrt{2}$ is irrational, that there is no surjective map $X\to2^X$, or that there are an infinity of primes, are certainly important or even "fundamental", but their proofs are so simple that one cannot call them "deep". A theorem is "deep" when its proof is really hard and, above all, requires a theory that transcends the realm the problem is formulated in. Consider, e.g., Gauss' theorem about which regular $n$-gons can be constructed with ruler and compass.

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    I think the theorem that $\sqrt{2}$ is irrational would have met your criterion for depth at the time it was discovered. Imagine that the Greeks saw $\sqrt{2}$ as the length of the diagonal of a unit square (rather than as the positive solution to $x^2=2$). Then to show the irrationality of $\sqrt{2}$, we need to transcend geometry to go to number theory, where we have available the fundamental theorem of arithmetic.2011-10-23
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    To add on Srivatsan's comment, the proof of $|P(X)|>|X|$ while seemingly trivial nowadays required the development of an entire new field in mathematics. I'd say this qualifies as pretty deep.2011-10-23