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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.

Could this be further generalized to n integers raised to nth power added together can have a number raised to nth power integer, hence why there is no relationship such that x^3 + y^3 = z^3, but there are an infinite number of x^3 + y^3 + z^3 = a^3 (where z in this case can work as +-1).

That's what I was asking :)

A conjecture related to this: https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_...

You know that it works for at least two values of w, but you don't know that it holds for any value of w.

The Pythagorean Theorem doesn't work for any arbitrary value of x, but there are an infinite set of x,y,z triplets that do work.