Case I: ( the case without solutions if you mean )

Note that: Just because

So: implies that and with

So:

And repeat the process until we get: which is a contradiction since !

Case II :Without loss of generality ( since we can exchange for freely) take

We have: and factorise

So we must have (since 2 is prime) : for and clearly since

Sum the 2 equations to get: , hence is even, and so

Thus it follows: (*) (solve the system)

Now take and (*) will produce solutions (note that hence can take 3 values )

Remember to consider the case which is totally analogous, all you have to do is to exchange for