<s>[INST] Find all integral solutions of the equation $x^n+y^n+z^n=2016$ 
Find all integral solutions of equation
  $$x^n+y^n+z^n=2016,$$
  where $x,y,z,n -$ integers and $n\ge 2$

My work so far:
1) $n=2$ $$x^2+y^2+z^2=2016$$
I used wolframalpha n=2 and I received the answer to the problem (Number of integer solutions: 144)
2) $n=3$ 
I used wolframalpha n=3 and I not received the answer to the problem 
How to do without wolframalpha?
 [/INST] An approach that can sometimes help to find solutions to this type of equation is to consider the prime factors of the given integer.  In this case:
$$2016 = 2^5.3^2.7 = 2^5(2^6-1) = 2^5((2.2^5)-1)$$
Hence:
$$\mathbf{2016 = 4^5 + 4^5 + (-2)^5}$$
And also (finding a solution of $x^3+y^3+z^3=252$ by trial and error or from this list):
$$2016 = 2^3.252 = 2^3(7^3 - 3^3 - 4^3)$$
Hence:
$$\mathbf{2016 = 14^3 + (-6)^3 + (-8)^3}$$
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