For $ n=0$ we have the solutions $ m=n=0$. For $ n\not=0$, we have no solution: since the factors in the product differ from eachother by a multiple of $ n$, they either all divide $ n$, or none of them does. In the first case we have $ \left|\prod (m+kn)\right > n^{2002}$, in the latter case we get a contradiction by looking at the equation modulo $ n$.

We may assume the gcd of m, n is 1 else we can divide it out. the equation mod n shows $m^{2002} = 0$ mod n $\rightarrow m = 0$ mod n. m = kn. ${n^{2002} = \prod_{j = 0}^2001}((k+j)n) \rightarrow \prod_{j = 0}^{2001}(k+j) = 1$ which is absurd.