Originally Posted by **malaygoel**

Here I am providing a very simple solution(only for natural numbers). If you find any mistake, tell me.

The solution goes like this:

3z^3 = x^3 +y^3 =(x+y)(x^2 – xy + y^2)

Now two things to note here:

(1)z>3(you can check it yourself)

(2)The first factor on the right hand side of the equation(x+y) is less than the second(x^2 – xy + y^2)(I can prove it to you if you want)

Now I will factorize 3z^3 into two factors with smaller first:

(1) 1,3z^3

(2) 3,z^3

(3) 3z,z^2

(4) z,3z^2

(Are there any more factors?)