So I am doing several questions about elementary number theory, and I am stuck in a question:
For positive integers $a,b,c$, prove that if $(ab+1)(bc+1)(ca+1)$ are perfect square, then $ab+1$, $bc+1$, $ca+1$ are all perfect squares
I have no idea on how to solve the problem. I try to assume $(ab+1)(bc+1)(ca+1)=(abc+k)^2$ and I get $$\dfrac{k^2-1}{abc}-\dfrac{1}{a}-\dfrac{1}{b}-\dfrac{1}{c}=a+b+c-2k$$ to make the right hand side a positive integer but it leads to nothing. Any ideas or solutions are appreciated, thanks.