How Many Good Pairs?

Number Theory Level 5

A pair of positive integers \((m, n)\) is called good if \(m\) divides \(n^2+1\) and \(n\) divides \(m^2+1.\)

A positive integer \(n\) is called admissible if there exists an integer \(m\) for which the pair \((m, n)\) is good.

Find the sum of all admissible positive integers \(\leq 100.\)

Details and assumptions
- As an explicit example, since \(5\) divides \(13^2+1\) and \(13\) divides \(5^2+1,\) the pair \((5, 13)\) is good. This also implies \(5\) are \(13\) are admissible integers.
- This problem is an extension of an old BMO problem.


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