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Feb16-12, 02:18 PM
Sci Advisor
HW Helper
P: 2,020
I'm gonna go ahead and guess that the problem of determining all integer solutions to
[tex]x^3+y^3+z^3+t^3=1999[/tex]is open.

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:
As of 2007 the only integers n such that [itex]0 \leq n \leq 100[/itex] and [itex]n \not\equiv \pm4 \, (\text{mod } 9)[/itex] that are not known to be a sum of three cubes are n=33, 42, 52, and 74.
This leads to me suspect that people don't know much about the rational solutions of this equation either (but I could be wrong).