It is not computationally tractable to enumerate all triples of integers up to the size required to find the values for 42.

So certainly any integer N either is or is not the sum of three cubes. Deciding which is the case computationally can be intractable.

Each of the integers in the solution are ~53 bits in length. To enumerate all triples of integers (with plus and minus) requires testing around 2^(53+53+53+1+1+1) = 2^162 items. If you could test one per nanosecond, this is 10^32 years, vastly older than the universe. If you threw 1 trillion such computers at it, it's still 10^23 years, also vastly longer than the age of the universe.

Thus enumeration is simply not possible.

The point of this is they found a solution for 42 after great effort.