Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2)+12$ is a prime number that properly divides the positive number $a^2+b^2+c^2+abc-2017$? My progress: Let $x=a-2$, $y=b-2$, $z=c-2$, then we get $xyz+12 \mid \left( x+y+z+4 \right)^2 - 45^2 \implies xyz+12 \mid (x+y+...