Find all $(a , b , c)$ such that $\frac{(a - b)(b - c)(c - a) + 4}{2}$ is power of $2016$. (Power of $2016$ means the number can be written in the form $2016^n$ for some non-negative $n$.

I don’t know what to do. Should I factorize $2016$ and consider by mod $32$ , mod $9$ and mod $7$?

Can anyone give me some hint please. Thank you!


Hints: First, let $x = b-a$ and $y = c-b$. Then, $(a-b)(b-c)(c-a) = xy(x+y)$.

We want $\dfrac{xy(x+y)+4}{2} = 2016^n$, i.e. $xy(x+y) = 2\cdot 2016^n-4$ for some integer $n \ge 0$.

If $n \ge 1$, then we need $xy(x+y) \equiv 5 \pmod{9}$. Is this possible? You have quite a few cases to check, but remember that none of $x$, $y$, and $x+y$ can be $0 \pmod{3}$.

If $n = 0$, then we need $xy(x+y) = -2$. This should be easy enough to solve.


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