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Three Years of Computer Time

No examples of other problems come to mind, but I have a speculation
as to why this kind of thing has traditionally been "recreational math
record-busting" and the like. What you are looking for is a problem
that takes a lot of computing power that (a) is significant enough
that someone cares about the answer but (b) noone cares enough to buy
supercomputer time to solve it quickly. (B) places an upper limit on
the degree of significance/cpu cycle, while (a) places a lower bound
on the total degree of significance. I suspect the band between these
two limits is narrow. I do think there are some interesting problems
in this domain, and I guess they would fall in the category of
propositions that are known for small n, and conjectured but not
proven or well explored for large n.

BTW, physicists can muster impressive computing equipment for
sufficiently interesting projects. In particular I recall an array of
several hundred MIPS R3000 CPUs to do gauge theory simulations.  There
was an Exobyte on every cluster (4-16 CPUs?)  to collect the data from
the simulation. Compared to an SSC, this requires only modest funding,
and it can be done with off-the-shelf hardware and a bit of
special-purpose software.