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wedrifid comments on What can you do with an Unfriendly AI? - Less Wrong
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As a non mathematician I am genuinely curious as to what we could do with a proof of the hypothesis that we could not do by merely assuming it. Since some consider it the most important problem in pure mathematics it must be something.
It has a bearing on encryption of course (given the relevance to prime numbers) however the proof would only make the encryption weaker so assuming it gives no vulnerability. It would seem from a naive viewpoint that more than the proof itself some of the intermediate steps required to construct the proof may be of greater potential value.
As is usual in mathematics, a proof of the Riemann hypothesis would almost certainly do more than just tell us that it's true- in order to prove something like that, you usually need to get a handle on deeper and more general structures.
For instance, Wiles' proof of Fermat's Last Theorem made massive progress on the modularity conjecture, which was then proved in full a few years later. Similarly, the proof of the Poincare conjecture consisted of a great and general advance in the "surgery" of singularities in the Ricci flow.
To think of an easier example, solving general cubic equations over the integers led mathematicians to complex numbers (it was easy to ignore complex solutions in the quadratic case, but in the cubic case you sometimes need to do complex arithmetic just to calculate the real root), and higher-order polynomial solutions led to Galois theory and more.
I once caught a nice popular talk by John H. Conway, about the sorts of extra things we would expect to learn from a proof of the Riemann Hypothesis.