Suppose, $m,k,c$ are positive integers
Conjecture : The expression $$m^2k^2(c^2+1)^2-4mc(c^2-c+1)$$ is a perfect square if and only if $m=k=1$
In the case $m=k=1$ , we get $(c-1)^4$ which is a perfect square ($0$ and $1$ are allowed)
The hard part is to show that otherwise the expression cannot be a perfect square. I tried to compare $(mk(c^2+1)\pm 1)^2$ with the given expression but this led to nowhere. The conjecture is true for $m,k,c\le 1\ 600$
I arrived at this problem by trying to prove that for positive integers $a,b,c$ with $c^2+1\mid a+b$ and $ab\mid c(c^2-c+1)$ we have $a=c$ , $b=c^2-c+1$ or vice versa.