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[Robert Gerbicz]

See http://euler.free.fr/
As you can see on their Resta page they've checked all numbers up to
700,000. ( using many modular tricks to speed up the search.)
This is a really hard problem, both in running time and memory. But there
are known solutions for n=7 and n=8, I don't know if these are the smallest
or not:
568^7=525^7+439^7+430^7+413^7+266^7+258^7+127^7  found by Mark Dodrill in
03/20/1999
1409^8=1324^8+1190^8+1088^8+748^8+524^8+478^8+223^8+90^8  found by Scott I.
Chase about in 2000.
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