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 lead to deep understanding of modularity of Galois representations.

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.

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 nonzero $n$th powers—but it lead to deep understanding of modularity of Galois representations.

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 lead to deep understanding of modularity of Galois representations.