It is given that $a,b,c \in \mathbb{Z}$
Let $(a-b) = -x$ and $(b-c)=-y$ then $(c-a) = x+y$ and $x,y \in \mathbb{Z}$
$(a-b)(b-c)(c-a) = xy(x+y)$
We can then distinguish between two cases:
Case 1: If $n=0$
$2016^n = 2016^0 = 1 \equiv 1 (\mod 2016)$
$xy(x+y) = -2$
$-2 = (-1)(-1)(+2) = (-1)(+1)(+2)=(+1)(+1)(-2)$
$(x,y) \in \{(-1,-1), (-2,1), (-1,2)\}$
$(a,b,c) = (k, k+1, k+2)$ and all cyclic permutations, with $k \in \mathbb{Z}$
Case 2: If $n>0$
$2016^n \equiv 0 (\mod 2016)$
$xy(x+y) + 4 = 2 \cdot 2016^n$
$2016 = 2^5 \cdot 3^2 \cdot 7$ is the unique prime factorization.
Using module arithmetic, it can be proved that there is no solution.
Method 1: Using modulo 9
$xy(x+y) + 4 \equiv 0 (\mod 9)$
$xy(x+y) \equiv 0 (\mod 9)$
In general, it is impossible to find integer values for $x,y$ to satisfy the last statement.
One way to show this is by checking all 27 possible cases modulo 9. An other one is the following:
$xy(x+y) + 4 \equiv 0 (\mod 3)$
$xy(x+y) \equiv 0 (\mod 3)$
$x \equiv 1 (\mod 3)$ and $y \equiv 1 (\mod 3)$
$x,y \in \{ 1(\mod 3), 4(\mod 3), 7(\mod 3) \}$
$xy(x+y) \equiv 2 (\mod 9)$ which is absurd since $xy(x+y) \equiv 5 (\mod 9)$ .
Method 2: Using modulo 7
$xy(x+y) + 4 \equiv 0 (\mod 7)$
$xy(x+y) \equiv 3 (\mod 7)$
In general, it is impossible to find integer values for $x,y$ to satisfy the last statement.
One way to show this is by checking all 15 possible cases modulo 7.