is a positive integer unique to the pair a,b except for exchanging a and b...

I tried proving it a few different ways but really I don't know why it seems to work. I looked at many examples where it holds and usually there are 3,6,9, or 12 solutions over the integers but except for a,b and b,a one of the two integers is always negative... Maybe the proof is simple, if 12 is the most solutions there can be and you know two solutions will both be negative integers, and 8 will have 1 negative and 1 positive integer, that only leaves two to be positive, but I wasn't able to get to that... Maybe also there's an argument from the symmetries that have to exist, I don't know...

For 3,4 the solutions are:

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