I was familiar with similar work using only positive integers (which of course makes the problem amenable to brute force approaches up to fairly large numbers), but I hadn't heard of this particular problem. I'm curious what practical implications (or applications) the knowledge (or proof of non-existence) for a some particular number has.
The goal (of a small number of mathematicians interested in that sort of problem) is to prove the hypothesis that all integers (except those congruent to 4 or 5 mod 9) can be written as a sum of three integer cubes. So in that sense this particular result has no value at all in achieving that goal. It might seem that it increases the probability of the hypothesis being true, but it really doesn't because it just demonstrates that one of an infinite number of integers satisfies the hypothesis.
So it's really just a bit of fun. Finding three-cube representations of specific integers doesn't help to prove the general case unless those representations reveal some kind of pattern which mathematicians can work with.
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