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?