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?



Why do we need to represent integers as the sum of three cubes?
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](https://arxiv.org/abs/2007.01209)*. 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.


