It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Some cases for integer $k$ becomes too hard like $42$ which it were presented as the following in 2019 by Bouker
$$
(−80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3=42,
$$ Some others $k$ less than $10^3$ are already known we cite $114,627,390,\ldots $. My question here is not to know how we can solve other cases for $k$, but my question is: what is the purpose behind the previous conjecture? Rather than that why do we need to represent integers as the sum of three cubes?
For example, spending a century of research to represent numbers like 114 or 390 or … as sum of three cubes is it just curiosity or will it provide a new addition to number theory?
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7$\begingroup$ Why not? We have considered many different problems of the sort "which integers can be represented in some particular form", and this is arguably the simplest one which is still open. $\endgroup$– WojowuCommented Dec 1, 2021 at 23:56
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6$\begingroup$ "is it just curiosity or will it provide a new addition to number theory?" It would be both! The very fact that 114 is representable would be a new addition to number theory, although that alone is merely a curiosity. What we are after is general methods which could tackle this problem in some generality and not just specific numbers. $\endgroup$– WojowuCommented Dec 2, 2021 at 0:00
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7$\begingroup$ Questions worded like this always rub me the wrong way, though I understand and appreciate the instinct behind them. I think the same question could be asked in a much less provocative way—even just saying "What are the implications of writing a number as a sum of three cubes?" could explore this territory without devaluing the work already put into it. (For example, does it matter that a specific number can or cannot be written as a sum of 3 cubes, or only whether or not all numbers can be so written? I don't know, and I think that would be an interesting question!) $\endgroup$– LSpiceCommented Dec 2, 2021 at 0:30
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1$\begingroup$ Very often trying to answer question like this leads to new results and tools (and some times to new areas) with applications to many other problems. Sometimes is the trip that is fascinating, not the destination. $\endgroup$– Nick SCommented Dec 2, 2021 at 1:50
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2$\begingroup$ Most mathematics is either trivial or hopelessly difficult. All the action lies in the tiny sliver of territory in between. The sum of three cubes lives in that sliver. We have a good grasp of quadratic equations, and a decent grasp of elliptic curves. The sum of three cubes is the next step up. Based on known technology, we have no reason to believe that $a^3 + b^3 + c^3 = k$ is unsolvable for $k\not\equiv \pm 4\pmod 9$, but we also can't prove it has solutions. The hope is that studying this equation will lead to new techniques, either of finding solutions or of proving unsolvability. $\endgroup$– Timothy ChowCommented Dec 2, 2021 at 15:59
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1 Answer
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The computational and theoretical number theory necessary to actually find such expressions is nontrivial and interesting. You should read the papers, eg Booker and Sutherland, On a question of Mordell. The actual problem is merely an excuse that drives people to develop new mathematics.
You might as well ask why we need to know that $n$th powers aren't the sum of two nonzero $n$th powers—but it led to deep understanding of modularity of Galois representations.
answered Dec 2, 2021 at 1:49