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From Scott Aaronson's blog:

There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof of the Riemann Hypothesis. Indeed, every formal proof of the Riemann Hypothesis with at most (say) a million symbols corresponds to some way of packing the boxes into your trunk, and vice versa. Furthermore, a list of the boxes and their dimensions can be feasibly written down.

His later commented to explain where he get this from: "3-dimensional bin-packing is NP-complete."

I don't see how these two are related.

Another question inspired by the same article is here.

Community
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Chao Xu
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1 Answers1

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The question of whether a formal proof of the Riemann Hypothesis exists (with at most a million symbols) is a problem in NP: given such a proof, it can be verified to be correct in polynomial time.

Bin-packing is NP-complete: this means that every problem in NP can be reduced to bin packing. In particular, the problem mentioned in the previous paragraph can. (This is a reduction that can be made explicit, so once we specify the proof verifier etc., we can carry out the steps of the reduction to get an instance of bin packing. We also need the reduction to be "parsimonious" i.e. solutions correspond one-to-one; I believe it is.)

ShreevatsaR
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  • The "solutions correspond one-to-one" (and onto) is key, or else his statement doesn't hold. I find that very hard to believe - reductions from A to B are rarely such that every problem in A corresponds to exactly one problem in B. – BlueRaja - Danny Pflughoeft Jul 28 '10 at 22:57
  • Problems don't have to correspond, only solutions for a fixed instance of each problem. I believe we do have such a "parsimonious" reduction (they're quite common actually), but even if we don't, something similar in spirit still holds: if we can solve bin packing, then we can solve it a few times to obtain a proof of the Riemann hypothesis. (E.g. solve the bin packing instances corresponding to "does there exist a proof with first bit 0?" then repeat once you know the first bit, etc.) – ShreevatsaR Jul 28 '10 at 23:33
  • so we are assuming any proof can be verified in polynomial time? If that's true then it all makes sense. – Chao Xu Jul 29 '10 at 01:49
  • @Chao Xu: Yes, a "formal proof" (see Wikipedia or this Notices of the AMS issue) is a proof written in a certain precise form that can be checked by computer. Actually, if we fix a bound on the proof length at 1 million, we don't even need to say "polynomial time", since the size is a constant. :-) – ShreevatsaR Jul 29 '10 at 01:59
  • See also this answer by T... – ShreevatsaR Aug 03 '10 at 05:14