I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is that a powerful and unexpected technique is introduced that comes to seem very natural once you are used to it.

**Example 1.** Euler's proof that there are infinitely many primes.

If you haven't seen anything like it before, the idea that you could use *analysis* to prove that there are infinitely many primes is completely unexpected. Once you've seen how it works, that's a different matter, and you are ready to contemplate trying to do all sorts of other things by developing the method.

**Example 2.** The use of *complex* analysis to establish the prime number theorem.

Even when you've seen Euler's argument, it still takes a leap to look at the complex numbers. (I'm not saying it can't be made to seem natural: with the help of Fourier analysis it can. Nevertheless, it is a good example of the introduction of a whole new way of thinking about certain questions.)

**Example 3.** Variational methods.

You can pick your favourite problem here: one good one is determining the shape of a heavy chain in equilibrium.  

**Example 4.** Erdős's lower bound for Ramsey numbers.

One of the very first results (Shannon's bound for the size of a separated subset of the discrete cube being another very early one) in probabilistic combinatorics.

**Example 5.** Roth's proof that a dense set of integers contains an arithmetic progression of length 3.

Historically this was by no means the first use of Fourier analysis in number theory. But it was the first application of Fourier analysis to number theory that I personally properly understood, and that completely changed my outlook on mathematics. So I count it as an example (because there exists a plausible fictional history of mathematics where it *was* the first use of Fourier analysis in number theory).

**Example 6.** Use of homotopy/homology to prove fixed-point theorems.

Once again, if you mount a direct attack on, say, the Brouwer fixed point theorem, you probably won't invent homology or homotopy (though you might do if you then spent a long time reflecting on your proof).

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The reason these proofs interest me is that they are the kinds of arguments where it is tempting to say that human intelligence was necessary for them to have been discovered. It would probably be possible in principle, if technically difficult, to teach a computer how to apply standard techniques, the familiar argument goes, but it takes a human to *invent* those techniques in the first place.

Now I don't buy that argument. I think that it is possible in principle, though technically difficult, for a computer to come up with radically new techniques. Indeed, I think I can give reasonably good [Just So Stories][1] for some of the examples above. So I'm looking for more examples. The best examples would be ones where a technique just seems to spring from nowhere -- ones where you're tempted to say, "A computer could never have come up with *that*."


  [1]: http://en.wikipedia.org/wiki/Just_So_Stories