Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is integer

Question

As stated in title, I would like to find solution to this problem:

Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is also integer.

I need idea how to solve this?

Assume without loss of generality that $x \leqslant y \leqslant z$. What is the largest that $x$ can be? For all $x$ not exceeding that bound, what is the largest that $y$ can be? — Daniel Fischer, Oct 09 '14 at 12:11

score 4 · Accepted Answer · answered Oct 09 '14 at 12:17

4

Let $1\leq x\leq y\leq z$ and $N= \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

If $N=3$ then $$x=y=z=1$$

If $N=2$ then $$ x=1,\ y=z=2$$

If $N=1$ then $$ 2\leq x\leq 3 $$

If $x=3$ then $$ (x,y,z)=(3,3,3) $$ If $x=2$ then $$ 3\leq y \leq 4$$

So $$ (x,y,z)=(2,3,6),\ (2,4,4)$$

answered Oct 09 '14 at 12:17

HK Lee

what about negative integers? – Leox Oct 09 '14 at 12:21
@Leox "Find all positive integers...". – AlexR Oct 09 '14 at 12:21
Ok, but in the first variant was negative. – Leox Oct 09 '14 at 12:25
In this case x=-y z=1 so there are infinitely many cases – HK Lee Oct 09 '14 at 12:39

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