Isn't the set of all triples of integers be enumerable, because (inductively) the set of all pairs of integers is enumerable (thanks, Cantor)?
Then, if one could enumerate all triples of integers, one could, for each triple, calculate the sum of the cubes.
So, integers which are sums of cubes are enumerable. That doesn't mean they're a known recursive set, just recursively enumerable.
Am I missing something? Perhaps the statement "The general problem of exactly which numbers are the sum of three cubes is unsolved." is meant to imply "not known to be a recursive set", i.e., no known algorithm for finite time answering the question of whether a given integer is in the set?
Then, if one could enumerate all triples of integers, one could, for each triple, calculate the sum of the cubes.
So, integers which are sums of cubes are enumerable. That doesn't mean they're a known recursive set, just recursively enumerable.
Am I missing something? Perhaps the statement "The general problem of exactly which numbers are the sum of three cubes is unsolved." is meant to imply "not known to be a recursive set", i.e., no known algorithm for finite time answering the question of whether a given integer is in the set?