A question on unit fractions 1/2+1/n = 1/x+1/y?

Interesting. We will acknowledge an infinitude of solutions (k,2,k), which are trivial.

Apart from that, I'm about to prove that you have already found ALL the solutions!

Let's then try to generate all solutions of the form (n,3,m), of which you already have two.

We require 1/m - 1/n = 1/6, so 1/m > 1/6 and you have apparently exhausted those.

Next let's go to solutions of the form (n,4,m);

we require 1/m - 1/n = 1/4, so 1/m > 1/4, but you've already exhausted that in your

slightly permuted solution (12,3,4).

Next, solutions of the form (n,5,m);

we require 1/m - 1/n = 3/10, so 1/m > 3/10 and can only be 1/3, already exhausted in your

slightly permuted solution (30,3,5).

Next, solutions of the form (n,6,m):

we require 1/m - 1/n = 1/3, so 1/m > 1/3, none available except 1/2.

Next, solutions of the form (n,7,m);

we require 1/m - 1/n = 5/14, so 1/m > 5/14, none available except 1/2.

Whenever that middle integer of the triple is greater than 6,

1/m - 1/n will be required to be greater than 1/3, so 1/m > 1/3,

and none will be available except 1/2.

You can now see that there are in fact NO solutions that differ in substance from

those that you've already given (except the trivial ones where m=n).