A number theory problem by Alejandro Castillo

Number Theory Level 5

\[\large \dfrac1m + \dfrac1n - \dfrac1{mn^2} = \dfrac34 \]

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