A number theory problem by Alejandro Castillo

1m+1n1mn2=34\large \dfrac1m + \dfrac1n - \dfrac1{mn^2} = \dfrac34

Let all the solutions of the pairs of integers (m,n)(m,n) that satisfy the equation above be denoted by (m1,n1),(m2,n2),,(mk,nk) (m_1, n_1), (m_2, n_2), \ldots, (m_k, n_k ) . Find i=1k(mi+ni)\displaystyle \sum_{i=1}^k (m_i + n_i) .

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