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42

 
Sheriff
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Many of you will remember from days past that "42" is the answer to the Great Question of Life, the Universe, and Everything.

But now two mathematicians have found another question to which 42 is the answer. You can read about it here: 42 is the answer to the question “what is (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³?”. You might not feel you're any good at math but you could read the article to see what mathematicians spend their time on.
 
Sheriff
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I'm confused by the article.  It states:

So numbers of the form 9𝑘+4 or 9𝑘+5 cannot be written as the sum of three cubes.


So let k = 1 and the numbers 13 and 14 can't be written as the sum of three cubes

But then it says

At that time, the only numbers less than 100 for which the problem was unsolved were 33, 42 and 74.


What about 13 and 14?  Were they solved or not?
 
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Yes, they were solved: they have no solution.
 
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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.
 
Paul Clapham
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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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