1979 AHSME Problem 22

Find the number of pairs (m, n) of integers which satisfy the equation m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1.

\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }3\qquad \textbf{(D) }9\qquad \textbf{(E) }\infty

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.