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At the present moment, what open mathematical problem do you seriously think you could solve if you had a very powerful computer at your disposition? I mean something like the Four-Color Problem, i.e. that is simple but more tractable with a brute-force approach than a "theoretical" one.

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I'm not sure there will be a lot of answers. For a professional mathematician working on a significant problem, I think it is generally not too hard to get access to serious computing resources. So problems that are tractable with such resources will most likely already have been solved. (Of course, this assumes that the resources required are realistic, which rules out things like computing Ramsey numbers.) –  Nate Eldredge Sep 13 '11 at 14:13
    
We all have very powerful computers at our disposal. With not too much money, you can start a thousand or ten thousand machines on Amazon Web Services or other cloud services. It's amazing to me. –  xpda Sep 13 '11 at 14:26
    
30,000 CPU linux cluster on Amazon Web Services: arstechnica.com/business/news/2011/09/… –  xpda Sep 20 '11 at 22:05
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For the last roughly 10 years, William Stein has been maintaining a powerful computer which he makes available to mathematical projects he considers worthwhile. There are 2 separate servers, each with 16 3GHz core processors, 128 GB RAM and 1.5 TB of hard disc; I don't know whether or not that counts as "super" in your estimation. Here are the specs he planned when he was building the current machine in 2008; I haven't been able to figure out whether it has been upgraded since. Earlier versions of this system go back to 2002-ish.

Here are the problems that he planned to have it work on back in 2008 and here is an update on what has been done so far. He has also convinced most of the researchers using his machine to make their directories world-readable, so you can actually watch ground breaking high-computational math in progress.

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Determining the existence of a projective plane of order 12. This is probably tractable with lots of moderately powerful computers, or even a few moderately powerful computers and lots of thinking (but we try to avoid that.)

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