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

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.