I was just trying to find all the integer triplets $(a,b,c)$ such that $ab+bc+ca=abc$. Actually this question popped up in my mind when I was just mathaholic.
By hit and trial I know that unordered triplet $(4,4,2)$ and $(0,0,0)$ satisfy the condition. But I couldn't find more.
I shall be thankful if you can give me some hint to prove that there exist infinite many such triplets (If really exist) otherwise if there are finite solutions then an algorithm to generate them all.
Thanks
If you divide both sides by $abc$, you end up with
$$\frac1a+\frac1b+\frac1c=1$$
Assuming they are positive integers greater than $0$ and $a\le b\le c$
Also assume $a=1$. Clearly, this is not possible.
Assume $a=2$. Some solutions involve $(b,c)=(4,4) $ and $(b,c)=(3,6)$.
Assume that $a=3$. The only solution here is $(b,c)=(3,3)$.
For $a>3$, there are no solutions.
If one includes negative integers, see that $(a,b,c)=(-n,1,n)$ is an infinite amount of such solutions.