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,\cdots $ , My question here is not to know how we can solve others 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 more detail 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?

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?

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,\cdots $ , My question here is not to know how we can solve others 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 ?

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,\cdots $ , My question here is not to know how we can solve others 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 more detail 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?