All solutions of $a+b+c=abc$ in natural numbers

I was observing some nice examples of equalities containing the numbers $1,2,3$ like $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$ and $\log 1+\log 2+ \log 3=\log (1+2+3)$. I found out this only happens because $1+2+3=1*2*3=6$.
I wanted to find other examples in small numbers, but I failed. How can we find all of the solutions of $a+b+c=abc$ in natural numbers?The question seemed easy, but it seems difficult to find. I would prefer an elementary way to find them!

What I did: We know if $a+b+c=abc$, $a|a+b+c$ so $a|b+c$. Similarly, $b|a+c$ and $c|a+b$.
Other than that, if we multiply both sides by $b$, we get $b^2+1=(bc-1)(ab-1)$.
If we also divide both sides by $abc$, we get $\frac{1}{bc}+\frac{1}{ac}+\frac{1}{ab}=1$.

I don't know how to go further using any of these, but I think they are a good start. I would appreciate any help.

Solution 1:

Without loss of generality $a \leq b \leq c$. Then $a+b+c \leq 3c$ and hence

$$abc=a+b+c \leq 3c$$

Thus, either $c =0$, in which case $a=b=c=0$, or

$$ab \leq 3 \,.$$

This leads to only four possibilities to check: $a=0$ or $(a,b)=(1,1)$ or $(a,b)=(1,2)$ or $(a,b)=(1,3)$.

Solution 2:

If $a=0$ then you require $b+c=0$ and hence $b=c=0$.

Note that you can assume $a\leq b \leq c$. If $a, b, c \geq 2$ then $abc \geq 4c > c + b + a$. Hence at least one of $a,b,c$ is equal to $1$.

Wlog assume $a=1$, and look for solutions to $b+c+1 = bc$. If $b,c\geq 3$ then $bc \geq 3c > b + c + 1$, hence at least one of $b,c$ is less than $3$

Wlog assume $b=2$, and look for solutions to $c+3 = 2c$, which implies $c=3$.

So the only solutions are $(0,0,0)$ and $(1,2,3)$ and their permutations.

Solution 3:

Here's a start of a full solution: The right side grows way faster than the left side, so it's unlikely that there are very many solutions. More formally, suppose that $a, b, c \ge 2$, and that $c$ is at least as large as $a, b$. Then we have

$$abc \ge 4c > c + c + c \ge c + b + a$$

so it's necessary that one of the numbers (which we'll call $a$) is $1$. So we can reduce the problem to studying

$$b + c = bc - 1$$

which has fewer variables.

Solution 4:

leaving aside the solution in which $a=b=c=0$ order the numbers so $a \le b \le c$

as OP shows, $c|(a+b)$ so we have:

$$ \frac{a+b}{c} \le 2 $$

thus only the values $1$ and $2$ are possible for $\frac{a+b}c$. these give $a+b=c$ and $a+b=2c$ respectively. however the latter is only possible if all three numbers are equal, as $c$ cannot be the arithmetic mean of two smaller numbers.

if all three numbers are equal they must each be $\sqrt{3}$, not a natural number solution.

plugging $a+b=c$ back into the equation $a+b+c=abc$ gives $ab=2$ hence $a=1, b=2, c=3$