Google and Duckduckgo use floating point rather than arbitrary precision integers. Wolfram Alpha's solver is significantly more advanced (including arbitrary precision for integers and floating point numbers)
Python's default number implementation has full support for arbitrarily large numbers, and will allocate more memory as needed to do so, making the original claim correct.
Maybe I'm looking at this the wrong way, but I'd handle enumerating this just like enumerating a set of three numbers. For an integer i just do (1-2(i%2)) * (i/2) using integer division. If it's recursively enumerable with natural numbers then it has to be trivially extendable to all integers just by that method alone.
> 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3