Sum of three cubes equal to a cube

Determine whether the Diophantine equation X^3 + Y^3 + Z^3 = T^3 has nontrivial integer solutions with |T| > |X| > |Y|, and, if they exist, characterize or construct such solutions.

Background

The paper discusses de Casteljau’s interest in the Diophantine equation X^3 + Y^3 + Z^3 = K and focuses on the special case K = T^3. Despite identities that produce sums of three cubes equalling specific integers, the general existence of nontrivial integer solutions when the right-hand side is itself a cube remains unresolved.

This problem is described as a still unsolved problem in number theory, distinguishing it from cases like Ramanujan’s taxicab identity where sums of two cubes coincide, and situating it within broader work on sums of three cubes.