Thread: I find this beautiful... View Single Post
$$x^3+y^3+z^3+t^3=1999$$is open.
$$x^3+y^3+z^3=1998=3^3\cdot 74$$in the integers. If we could solve that, then we'd know a thing or two about the rational solutions to
$$X^3 + Y^3 + Z^3 = 74.$$ 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 $0 \leq n \leq 100$ and $n \not\equiv \pm4 \, (\text{mod } 9)$ that are not known to be a sum of three cubes are n=33, 42, 52, and 74.