Well I was doing some questions and i found something. This equation $x^3+y^3+z^3=w^3$ has only one solution which is $x=3,y=4,z=5,w=6$.

And what I have have proposed is that there is not other natural number solution for all values of $x,y,z,w>1$. So my question is this proposition true and if it is can we prove that there is only one natural number solution to this.

well there is a condition for this that is these numbers add up to give the cube of a perfect number.

infinitely manyinteger solutions of the form $(x, \, -x, \, z, \, z)$. $\endgroup$ – Francesco Polizzi Jan 22 '15 at 10:38