As a first attempt, I guess you'd create a table of 5th powers of integers through (say) 1000, then look for combinations of two, three, or four entries from the same array that added up to one of the same numbers. Or more likely, walk through the array and subtract all combinations of one, two, three, or four elements from each, looking for zeroes. The most naive search through such a table could take on the order of a trillion trials to find a counterexample, which is pretty painful, but I'm sure there are plenty of ways to optimize it. Especially if you worked from high to low, you could skip enormous chunks of the search space that would take the computation out of range.

The CDC 6600 was a pretty hardcore machine; it was Seymour Cray's first design at his Chippewa Falls skunk works, according to Wikipedia. Would definitely be interesting to know how long their solution took.

There's a more detailed discussion at "A Survey of Equal Sums and Power" by L. J. Lander, T. R. Parkin and J. L. Selfridge http://www.jstor.org/stable/2003249 (under 5.1.n)

they probably used the meet in the middle technique: http://www.infoarena.ro/blog/meet-in-the-middle

One could probably start with the fact that the last 2 digits of any 5th power are in "0, 1, 7, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99" and go from there.