Read Carefully

Obviously this problem is harder than just factoring integers, so if you can't factor, you can't find all soultions.

The real problem would be to determine if there exist solutions for a pair M and N.

To find a,b,c and d is harder than factoring.
But there also doesn't seem to be some way to easily know if there is a solution, and I doubt there is.

As Gauss said after looking at Fermat's Last Theorem, anyone can put infinitely many problems which are easy to state, hard to prove, and most importantly - meaningless.
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