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> Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$. What are the possible values of $$\frac{x^2+y^2-1}{xy}$$?
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I have discovered infinitely many pairs. They are $(n, n+1)$. This solution works for all $n$. For all of them the value of $$\frac{x^2+y^2-1}{xy}=2$$

But I want to know if there are any other such solutions. Also what are all the possible values. I have discovered 2. There may be many such values. Are they finite? Can we prove their finite or infinite existence? 

Any help would be appreciated.