A rectangular prism has **integer** edge lengths. Find all dimensions such that its surface area equals its volume.

My Solution:

Let the edge lengths be represented by the variables $l, w, h$

Then $$lwh = 2\,(lw +lh + wh) \implies lwh = 2lwh\left(\frac{1}{h} + \frac{1}{w} + \frac{1}{l}\right)$$

Dividing both sides of the equation by $lwh$ yields $$1 = 2\left(\frac{1}{h} + \frac{1}{w} + \frac{1}{l}\right)$$

Or, $$\frac{1}{l} + \frac{1}{w} + \frac{1}{h} = \frac{1}{2}$$

Though perhaps a bit unnecessary, I used some algebraic deduction and number theory to find all the possible ordered triple pairs for the dimensions of the rectangular prism in the cases where **all** the dimensions are the same and **two** of the dimensions are the same. 

My answers were: $(6,6,6),(5,5,10),(8,8,4),(12,12,3)$

I have a hunch that no ordered pair exists where all three values are distinct, but is there a way to rigorously prove this?

Note: By the AM-GM Inequality, $lwh \geq 216$. (I haven't been able to make use of this fact, but I just noted it here in case)

**Edit**: After researching a little more about Egyptian Fraction Analysis, inspired by marty cohen's answer, I found from http://mathworld.wolfram.com/EgyptianFraction.html that a unit fraction can be split into two more unit fractions: $$\frac{1}{a} = \frac{1}{a+1} + \frac{1}{a(a+1)}.$$
I then used this idea to successfully generate a few ordered triplet solutions ($l,w,h$). 

WLoG, assume $l \leq w \leq h$. Then the following ordered triplets ($l,w,h$) are solutions of the equation $\frac{1}{l} + \frac{1}{w} + \frac{1}{h} = \frac{1}{2}$. 

$$(4,6,12), (3,7,42), (3,8,24)$$

But, I suspect there are many more solutions, and I don't know how to use casework or an otherwise more organized approach to solve this problem. I do believe it can be solved using the idea of splitting a unit fraction into two other unit fractions, but the possibilities seem endless.

If anybody has any idea on how to solve the problem using the "splitting unit fraction" method, or has any other working method, I would greatly appreciate it if you shared it with me.

Thanks

-A