I'm gonna go ahead and guess that the problem of determining all integer solutions to
I have two reasons to suspect this: 1) These kinds of problems are generally really, really, really
difficult to solve (cf. FLT, Erdos-Strauss, ...). 2) Let's suppose we were able to classify all integer solutions. We would then, in particular, be able to classify integer solutions in which t=1, or equivalently, we'd be able to solve
[tex]x^3+y^3+z^3=1998=3^3\cdot 74[/tex]in the integers. If we could solve that, then we'd know a thing or two about the rational solutions to
[tex]X^3 + Y^3 + Z^3 = 74.[/tex] I believe nobody knows if this last equation has any integer solutions. Indeed, in Cohen, Number Theory: Analytic and Modern Tools
(Springer, 2007), one finds the following piece of info:
This leads to me suspect that people don't know much about the rational solutions of this equation either (but I could be wrong).