In fact every integer k is the sum of nine cubes, and every multiple of 6 is the sum of four cubes. Your equation is just the equation x^3 + y^3 + (-z)^3 + w^3 = 6k where k = 0.
The more general problem is known as Waring's problem. For example, every integer is the sum of 19 fourth powers. Of course special families of integers require fewer fourth powers.
The more general problem is known as Waring's problem. For example, every integer is the sum of 19 fourth powers. Of course special families of integers require fewer fourth powers.