TITLE: Proving the Riemann Hypothesis without revealing anything other than you proved it
QUESTION [13 upvotes]: Consider the following assertion from Scott Aaronson's blog:

Supposing you do prove the Riemann
  Hypothesis, it’s possible to convince
  someone of that fact, without
  revealing anything other than the fact
  that you proved it. It’s also possible
  to write the proof down in such a way
  that someone else could verify it,
  with very high confidence, having only
  seen 10 or 20 bits of the proof.

Can anyone explain where this result comes from?

REPLY [10 votes]: The point is that SHORTPROOF is an NP-complete problem: given a sentence of length $n$ in the language of some formal proof system (ZFC, Peano arithmetic, etc), does it have a proof of length at most some fixed polynomial in $n$, such as $(2n)^{100}$?  It's in NP because for a reasonable formal system you can check a given proof fairly quickly.  This problem was considered in Goedel's letter to von Neumann that implicitly stated what we now call the $P \neq NP$ question (the heart of the problem, the universality of concrete NP-complete problems, wasn't known until much later). 
Any NP-complete problem has a zero-knowledge proof protocol for demonstrating solutions of instances, e.g. that we have a SHORTPROOF of the Riemann hypothesis.  These are "proofs that reveal nothing other than their own validity".
The role of the PCP theorem is to show that the proof protocols (interactive challenge/response games) can be very efficient for any stipulated level of confidence.  The probability that the prover really does have a SHORTPROOF of Riemann, given that we follow the protocol and the prover wins, is at least 99 percent, or whatever specified degree of certainty.