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The title is taken from the movie "The Prestige" (very good, by the way).

To explain it, let me quote Michael Caine in the film:

Every great magic trick consists of three acts. The first act is called "The Pledge"; The magician shows you something ordinary, but of course... it probably isn't. The second act is called "The Turn"; The magician makes his ordinary some thing do something extraordinary. Now if you're looking for the secret... you won't find it, that's why there's a third act called, "The Prestige"; this is the part with the twists and turns, where lives hang in the balance, and you see something shocking you've never seen before.

Maybe it's a bit of a stretch, but I think that some series of mathematical statements have this structure. The pledge would be a set of hypotheses. The turn would be a perfectly fine first theorem based on the pledge. The prestige would be the very surprising second theorem that would make you reconsider the whole thing, and certainly feel like "you see something shocking you've never seen before".

I have two examples.

  • Goodstein's theorem, favorited by most as a "theorem with unexpected conclusions" in the Theorems with unexpected conclusions thread (see the explanation there, it's truly beautiful).
  • The second statement on that same list of "theorems with unexpected conclusions". Actually, the theorem by Elkies presented there would be the prestige, and the turn is that the only Fibonacci cubes are $0,\pm 1,\pm 8$ (i.e. Is 8 the largest cube in fibonacci sequence? )

This is of course the motivation to ask: our two most favorited unexpected theorems have the same two-punch structure, and maybe we could find more if we look for that structure only?

To qualify it a bit more, I would like to have two theorems in a row. The first, while a perfectly legitimate result on its own, would only feel after seeing the second theorem completely different, as a general misdirection. Yet that first result would be needed on the way to prove the second result, much deeper or of an unusual form. Certainly this relation between the two theorems would be unexpected.

So to truly ask a question: can you think of more examples of a "prestige" in mathematics?

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closed as not a real question by Franz Lemmermeyer, Loop Space, HJRW, Robin Chapman, Cam McLeman Sep 28 '10 at 18:00

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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I don't see a mathematical question here. –  Loop Space Sep 28 '10 at 17:07
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@ anonymous: please try to make your question a bit more precise. I understand the way you'd like to pose your question, but it would yield (more) interesting answers if you would define your question better. Perhaps someting like: "What are some examples of mathematical theorems that where found by means of an interesting proof, which at first sight had nothing to do with the result until you read the last line" comes close to your line of thought? I really enjoyed watching that Film too, by the way. –  Max Muller Sep 28 '10 at 17:33
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I think is a fun, well-thought-out, well-written question which is nonetheless not really appropriate for this site. No downvote, but voting to close. –  Cam McLeman Sep 28 '10 at 18:00

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