Find all positive integer solutions to a + b + c = a * b * c

[jobyts]

a + b + c = a * b * c

where a, b, c are positive integers.

I can think of only one solution to this. {1, 2, 3}.

Is there any other solution to it?
Can you prove or disprove?

 

[Office_Shredder]

Without loss of generality assume $a\geq b \geq c$. If $b\geq c\geq 2$ then
$a*b*c \geq 4a$
Which is necessarily larger than $a+b+c \leq 3a$.

So c=1 necessarily. Then we have
$a+b+1 = a*b$
Now assume that b>2. The right hand side is at least 3a, and the left hand side is smaller than 2a+1, and we know that a is larger than 1 so these two cannot be equal. Therefore b=1 or b=2

If b=1 and c=1 there is obviously no solution (we get a+2 = a). If c=1 and b=2 we get a+3 = 2a which is solved by a=3. So {1,2,3} is the only positive integer solution.

 

[jobyts]

Thank you.