$m^3 - n^3 = 7mn + 5 \Rightarrow m \geq n, (m - n)(m^2 + mn + n^2) = 7mn + 5 \Rightarrow $ $ (m - n)((m - n)^2 + 3mn) = (m - n)^3 + 3(m - n)mn = 5 + 7mn$ Here we take it by cases Case 1: $3(m - n) \geq 7 \Rightarrow m - n \geq 3 \Rightarrow (m - n)^3 \geq 27 \Rightarrow$ $(m - n)^3 + 3(m - n)mn > 5 + 7mn $. So no solutions. Case 2: $3(m - n) < 7 \Rightarrow m - n \leq 2 \Rightarrow m - n = 1, 2$ If $m - n = 1$ L.H.S $<$ R.H.S $\Rightarrow m - n = 2$ we get $8 + 3(2)mn = 5 + 7mn \Rightarrow 8 + 6mn = 5 + 7mn \Rightarrow 3 = mn$ and $m - n = 2 \Rightarrow m = 3, n = 1 \Rightarrow (m,n) = (3,1)$ are the only solutions.