$2 \times 3 = 5+1$ and $2+3 = 5 \times 1$. When else can we switch the operators like this?

Question

I noticed the following: $$2 \times 3 = 5+1$$ If you switch the operators, it is still true: $$2+3 = 5 \times 1$$ There is another obvious/trivial example where you can swap the operators: $$2\times 2 = 2+2$$ I think these are the only solutions (for positive integers). Can anyone give an elegant proof?

Answer 1

Your claim is correct. Indeed, suppose that $ab = c+d$ and $a+b = cd$. It is clear that at most one of the values $a,b,c,d$ can be equal to $1$, so w.l.o.g. we may assume that $a$ and $b$ are both at least $2$.

We have $$ab = c + \frac{a+b}{c} \leq (a+b)+1,$$ and dividing by $b$ gives $$a \leq \frac{a}{b} + 1 + \frac{1}{b} < \frac{a}{2} +2,$$ from which we get $a < 4$; hence $a=2$ or $a=3$. Similarly, $b=2$ or $b=3$. The four remaining possibilities for $(a,b)$ can now be checked one by one.