In 1966, in a famous two-sentence paper, they announced that 27^5 + 84^5 + 110^5 + 133^5 equaled 144^5, the first counterexample to the Euler Conjecture. They mentioned the Control Data 6600, but nothing else. How would they have gone about hunting those numbers up? Are there still only a couple other known fifth-power counterexamples? Do people hunt for them, or hunt for a method to generate them?
It wouldn’t take much in efficiency gains to make a mostly-brute-force search viable, even with the computing power of the time. And I’m sure you could get some nice efficiency gains from modular arithmetic.
Naturally people still search for similar identities. There is a database of hundreds of examples here, although it seems that it hasn’t been updated for a few years. That page has the similar example
85282^5+28969^5+3183^5+55^5 = 85359^5
ETA: parlty ninja-ed by @markn_1.
This Euler’s sum of powers conjecture / counterexamples - Wikipedia suggests me at least that there was a bit of good luck.
The other known counterexamples to the k = 5 case involve vastly larger numbers that would have been very hard to discover by any search algorithm on the hardware of that era. But the comparatively small counterexample with factors just barely over 100 existed to be found. Which they duly did. And all congratulations to them for that.
A table of the first e.g. 200 5th powers is still just 200 numbers. Which could be combined using a reasonably smart, but still brute force, algorithm examining 200^2 possibilities for n=2, 200^3 for n=3, and 200^4 for n=4. Those are not huge numbers of iterations even for ancient hardware.
If they’d had to brute force their way to the next known counterexample, where they’d be exploring combinations from a table of ~15,000 or ~85,000 possible factors, they’d probably have run out of horsepower real quickly.